<!DOCTYPE html><html lang="en">
      
      
        
    
    
    
    

      
      
      <head>
        <meta charset="utf-8">
        <meta name="format-detection" content="telephone=no">  
        
        <title>Lecture 13: Sums and Asymptotics | Video Lectures | Mathematics for Computer Science | Electrical Engineering and Computer Science | MIT OpenCourseWare</title>
    <!-- Begin Automatic Metadata Insertion -->
    <meta content="6-042j-mathematics-for-computer-science-fall-2010" name="WT.cg_n">
    <meta content="Lecture 13: Sums and Asymptotics" name="WT.cg_s">
    <meta content="Analysis of sums, formulation of asymptotic bounds using various techniques, and introduction to asymptotic notation." name="Description">
    <meta content="Leighton, Tom" name="Author">
    <meta content="Dijk, Marten van " name="Author">
    <meta content="6.042J,6.042,18.062J,18.062,block stacking,greedy algorithm,center of mass,sum,recursion,harmonic numbers,closed form expression,Sterling's Formula,bounds,Computer Science,Probability and Statistics,Applied Mathematics,Discrete Mathematics" name="keywords">
    <meta content="6.042J Mathematics for Computer Science | Lecture 13: Sums and Asymptotics" name="Search_Display">
    <meta content="Computer Science" itemprop="about">
    <meta content="Probability and Statistics" itemprop="about">
    <meta content="Applied Mathematics" itemprop="about">
    <meta content="Discrete Mathematics" itemprop="about">
    <!-- End Automatic Metadata Insertion -->

	<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/grid.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/base.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/menu.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/jquery.bubblepopup.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/search.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/courses.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/courses_new.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/jquery.jscrollpane.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/media_tabs.css">
	<link href="../../../common/xml/ocwcc.rdf" type="application/rdf+xml" rel="metadata">
	<link rel="canonical" href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-13-sums-and-asymptotics/">
	<link rel="apple-touch-icon" href="../../../common/images/apple-touch-icon.png">
	
	
      
	
	<script type="text/javascript" src="../../../common/scripts/jquery.js"></script>
	<script type="text/javascript" src="../../../common/scripts/ocw-media-utils-offline.js"></script>
	<script type="text/javascript" src="../../../common/scripts/ocw-offline.js"></script>
	<script type="text/javascript" src="../../../common/scripts/jquery.bubblepopup.min.js"></script>
	<script type="text/javascript" src="../../../common/scripts/jquery-ui.min.js"></script>
	<script type="text/javascript" src="../../../common/scripts/jquery.jscrollpane.min.js"></script>
	<script type="text/javascript" src="../../../common/scripts/expandy.js"></script>
	<script type="text/javascript" src="../../../common/scripts/bubble-popup-offline.js"></script>
	
	
	
    <script type="text/javascript">
      $(document).ready(function() {
        $("#tabs").tabs();
        IpadScroller();
      });
    </script>
    
    
    
    
    
    
      
       
		 

        
        
        

        
        
        
        
        
        
        
        
        
      </head>
    <body itemscope itemtype="http://schema.org/WebPage">
        
	

        <header id="top">
			<div id="grid">
				
				
					
<div id="portletwrapper-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d686561646572" class="portletWrapper kssattr-portlethash-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d686561646572">
<div class="portletStaticText portlet-static-site-header">
<!--googleoff: index-->
<div class="grid_6 alpha" id="banner"><a href="https://ocw.mit.edu/"><img src="../../../common/images/ocw_mast.png" class="logo" alt="MIT OpenCourseWare, Massachusetts Institute of Technology"></a></div>
<div class="grid_6 omega" id="subscribe">
<aside class="module" aria-label="Connect with OCW">
<table class="social">
    <tbody>
        <tr>
            <td class="socialbutton"><a aria-label="Subscribe to the OCW Newsletter" href="https://ocw.mit.edu/subscribe/index.htm?utm_source=header"><img src="../../../common/images/trans.gif" alt="An icon depicting an envelope.">Subscribe to the OCW Newsletter</a></td>
            <td>
<a aria-label="Facebook" href="https://facebook.com/mitocw"><img src="../../../common/images/icon_fb.png" alt="Click to visit our Facebook page."></a>  <a aria-label="Instagram" href="https://www.instagram.com/mitocw/"><img src="https://ocw.mit.edu/images/icon_ig.png" alt="Click to visit our Instagram page."></a> <a aria-label="Twitter" href="https://twitter.com/mitocw"><img src="https://ocw.mit.edu/images/icon_twitter.png" alt="Click to visit our Twitter feed."></a><a aria-label="YouTube" href="https://www.youtube.com/mitocw" style="font-size: 12.208px;"><img src="https://ocw.mit.edu/images/icon_yt.png" alt="Click to visit our YouTube page."></a><span style="font-size: 12.208px;">   </span><a aria-label="Open Matters blog on WordPress" href="https://www.ocw-openmatters.org/" style="font-size: 12.208px;"><img src="https://ocw.mit.edu/images/icon_wp.png" alt="Click to visit our blog on WordPress."></a>
</td>
        </tr>
    </tbody>
</table>
</aside><nav aria-label="Help Links" class="helplinks">     <a aria-label="OCW Site Help" href="https://ocw.mit.edu/help">Help</a><span aria-hidden="true">|</span>     <a href="../../../common/about/contactus.htm">Contact Us</a>   </nav>
</div>
<div class="clear"> </div>
<!--googleon: index-->
</div>

</div>

<div id="portletwrapper-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a6d6567612d6d656e75" class="portletWrapper kssattr-portlethash-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a6d6567612d6d656e75">
<div class="portletStaticText portlet-static-mega-menu"><div><nav id="mega" class="grid_8 alpha" aria-label="Site">
<ul id="menu" role="presentation">
    <li id="menu_home">
<a href="https://ocw.mit.edu/" aria-label="Homepage"><img src="../../../common/images/top-nav_home.png" class="home_icon" alt="Click for site home page."></a><!-- Begin Home Item -->
</li>
    <!-- End Home Item -->
    <li id="drop_1" aria-label="Find Courses">
<a href="#" aria-hidden="true">FIND COURSES</a><!-- Begin 5 columns Item -->
    <div class="dropdown_5columns-a mega-courses">
    <div class="col_1a">
    <div class="row_1a">
<nav aria-labelledby="mm-find-courses-by">     <span id="mm-find-courses-by" class="nav" aria-hidden="true">Find courses by:</span>
    <ul class="find_by" role="presentation">
        <li><a href="https://ocw.mit.edu/courses/find-by-topic/">Topic</a></li>
        <li><a href="https://ocw.mit.edu/courses/find-by-number/">MIT Course Number</a></li>
        <li><a href="https://ocw.mit.edu/courses/find-by-department/">Department</a></li>
    </ul>
    </nav>     <nav aria-labelledby="mm-collections">     <span id="mm-collections" class="nav" aria-hidden="true">Collections</span>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/new-courses/">New Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/most-visited-courses/">Most Visited Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/ocw-scholar/">OCW Scholar Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/audio-video-courses/">Audio/Video Lectures</a></li>
        <li><a href="https://ocw.mit.edu/courses/online-textbooks/">Online Textbooks</a></li>
        <li><a href="https://ocw.mit.edu/resources/">Supplemental Resources</a></li>
        <li><a href="https://ocw.mit.edu/high-school/">OCW Highlights for High School</a></li>
        <li><a href="https://ocw.mit.edu/courses/mitx-related-courseware/">MITx &amp; Related OCW Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/mit-open-learning-library/">MIT Open Learning Library</a></li>
    </ul>
    </nav>     <nav class="col_1b" aria-labelledby="mm-translated-courses">     <span id="mm-translated-courses" class="nav" aria-hidden="true" style="line-height: 1.3;">Cross-Disciplinary Topic Lists</span>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/energy-courses">Energy</a></li>
        <li><a href="https://ocw.mit.edu/courses/entrepreneurship">Entrepreneurship</a></li>
        <li><a href="https://ocw.mit.edu/courses/environment-courses">Environment</a></li>
        <li><a href="https://ocw.mit.edu/courses/intro-programming">Introductory Programming</a></li>
        <li><a href="https://ocw.mit.edu/courses/life-sciences">Life Sciences</a></li>
        <li><a href="https://ocw.mit.edu/courses/transportation-courses">Transportation</a></li>
    </ul>
    </nav>
</div>
    <div class="row_1b"><nav aria-labelledby="mm-cross-disciplinary-topic-lists">     <span id="mm-cross-disciplinary-topic-lists" class="nav" aria-hidden="true">Translated Courses</span>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/translated-courses/traditional-chinese" aria-label="Traditional Chinese">繁體字 / Traditional Chinese</a></li>
        <li><a href="https://ocw.mit.edu/courses/translated-courses/turkish" aria-label="Turkish">Türkçe / Turkish</a></li>
        <li><a href="https://ocw.mit.edu/courses/translated-courses/korean" aria-label="Korean">(비디오)한국 / Korean</a></li>
    </ul>
    </nav></div>
    </div>
    </div>
    </li>
    <li id="drop_2">
<a href="#" aria-label="For Educators">For Educators</a>
    <div class="dropdown_1column-a" style="width: 270px;"><nav aria-labelledby="mm-find-courses-by">
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/educator/chalk-radio-podcast">Chalk Radio Podcast</a></li>
        <li><a href="https://ocw.mit.edu/educator/">OCW Educator Portal</a></li>
    </ul>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/instructor-insights/">Instructor Insights by Department</a></li>
        <li><a href="https://openlearning.mit.edu/campus/digital-innovations/">Residential Digital Innovations </a></li>
    </ul>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/high-school/">OCW Highlights for High School</a></li>
    </ul>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/educator/additional-resources/">Additional Resources</a></li>
    </ul>
    </nav></div>
    </li>
    <li id="drop_3">
<a href="#" aria-hidden="true">Give Now</a>
    <div class="dropdown_1column-a"><nav class="col_1" aria-label="Donate">
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/give/">Make a Donation</a></li>
        <li><a href="https://ocw.mit.edu/give/why-give/">Why Give?</a></li>
        <li><a href="https://ocw.mit.edu/give/our-supporters/">Our Supporters</a></li>
        <li><a href="https://ocw.mit.edu/give/other-ways-to-contribute/">Other Ways to Contribute</a></li>
        <li><a href="https://ocw.mit.edu/support/">Become a Corporate Sponsor</a></li>
    </ul>
    </nav></div>
    </li>
    <li id="drop_4">
<a href="#" aria-hidden="true">About</a>
    <div class="dropdown_1column-a"><nav class="col_1" aria-label="About">
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/about/">About MIT OpenCourseWare</a></li>
        <li><a href="https://ocw.mit.edu/about/site-statistics/">Site Statistics</a></li>
        <li><a href="https://ocw.mit.edu/about/ocw-stories/">OCW Stories</a></li>
        <li><a href="https://ocw.mit.edu/about/newsletter/">Newsletter</a></li>
        <li><a href="https://chalk-radio.simplecast.com/">Chalk Radio Podcast</a></li>
        <li><a href="https://www.ocw-openmatters.org/">Open Matters Blog</a></li>
    </ul>
    </nav></div>
    </li>
</ul>
</nav></div></div>

</div>





<!--googleoff: index-->
<script>
  (function() {
	var cx = '012626166551961672889:owjdpuboktq';
	var gcse = document.createElement('script');
	gcse.type = 'text/javascript';
	gcse.async = true;
	gcse.src = 'https://cse.google.com/cse.js?cx=' + cx;
	var s = document.getElementsByTagName('script')[0];
	s.parentNode.insertBefore(gcse, s);
  })();
  window.onload = function(){
	document.getElementById('gsc-i-id1').placeholder = 'Search';	
  };

$(document).ready(function(){

$('.advanceSearch a').keydown(function(event){showSearchTips($(this),event);})
$('#searchTipsModal').keydown(function(event){showSearchTips($(this),event);})

function showSearchTips(obj,evt) {
// if pressed enter key
	if ( evt.which == 13 || evt.which == 32) {
		showModal();
		ga('send', 'pageview', "AdvanceSearch");
		$(".advanceSearch a").attr('aria-expanded', 'true');
		$('.searchTipsModal').focus();
		evt.preventDefault();
		}
	if ( evt.which == 27 ) {
		hideModal();
		$(".advanceSearch a").attr('aria-expanded', 'false');
		var modal = document.getElementById('searchTipsModal');
		modal.style.display = "none";
		$(".advanceSearch a").focus();
		evt.preventDefault();
		}
}

});
function showModal(){
	var modal = document.getElementById('searchTipsModal');
	modal.style.display = "block";
	ga('send', 'pageview', "AdvanceSearch");
	document.getElementById("searchTipsBtn").setAttribute("aria-expanded", true);
	window.setTimeout(function () {
    document.getElementById('searchTipsModal').focus(); }, 0);
}

function hideModal(){
	var modal = document.getElementById('searchTipsModal');
	modal.style.display = "none";
	document.getElementById("searchTipsBtn").setAttribute("aria-expanded", false);
}
</script>
<div id="search" role="search" class="grid_4 omega">
    	<table class="search">
				<tbody>
					<tr>
						<td><div class="searchboxheader"><searchbox-only resultsurl="/search/ocwsearch.htm"></searchbox-only></div></td>
						<td>
							<div class="advanceSearch">
								<a id="searchTipsBtn" onclick="showModal();" href="#" role="button" aria-label="search tips" aria-expanded="false" aria-describedby="searchtips">Search Tips</a>

								<!-- The Modal -->
								<div id="searchTipsModal" class="modal" tabindex="-1">
								  <!-- Modal content -->
									<div class="modal-content">
										<div class="modal-body">
											<button class="close" onclick="hideModal();" aria-label="close">X</button>
											<span>
												<b>Exclude words from your search</b>
												<br>Put - in front of a word you want to leave out. For example, jaguar speed -car
												<br><br>
												<b>Search for an exact match</b>
												<br>Put a word or phrase inside quotes. For example, "tallest building".
												<br><br>
												<b>Search for wildcards or unknown words</b>
												<br>Put a * in your word or phrase where you want to leave a placeholder. For example, "largest * in the world".
												<br><br>
												<b>Search within a range of numbers</b>
												<br>Put .. between two numbers. For example, camera $50..$100.
												<br><br>
												<b>Combine searches</b>
												<br>Put "OR" between each search query. For example,  marathon OR race.
												<br><br>
											</span>
										</div>
								  </div>
								</div>
							</div>
						</td>
					</tr>
				</tbody>
		</table>
</div>
<div class="clear"></div>
<!--googleon: index-->
<!-- *end header* -->

				
				
			</div>
<!-- top grid end -->
		</header><!-- top end -->
			
		<div id="center_media">
      	<div id="grid">
      		<div id="left">
        		<nav id="breadcrumb_media" aria-label="Breadcrumb">
                	<p>

    <a href="https://ocw.mit.edu/">Home</a>
    
        »
        
    
    
        
            <a href="https://ocw.mit.edu/courses">Courses</a>
            
                »
                
            
            
         
    
    
        
            <a href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science">Electrical Engineering and Computer Science</a>
            
                »
                
            
            
         
    
    
        
            <a href="../../../contents/index.htm">Mathematics for Computer Science</a>
            
                »
                
            
            
         
    
    
        
            <a href="../../../contents/video-lectures/index.htm">Video Lectures</a>
            
                »
                
            
            
         
    
    
        
            
            
            Lecture 13: Sums and Asymptotics
         
    
</p>

            	</nav>
            	<div class="clear"></div>
        		<div id="media_title">
        		<h1 class="title" itemprop="name" property="dct:title">
        <span class="" id="parent-fieldname-title">
            Lecture 13: Sums and Asymptotics
        </span>
    </h1>
        		</div>
           		<div class="clear"></div>
           		<div id="course_wrapper_media">
           			<nav id="course_nav" aria-label="Course">
           				<script language="javascript" type="text/javascript">
function toggleMenu(objID) {
  if (!document.getElementById) return;
  var ob = document.getElementById(objID);
  ob.className = (ob.className == 'selected')?'': 'selected';
}
function toggleClass(id)
{
  var divtoggleClass= document.getElementById(id);
  divtoggleClass.className = (divtoggleClass.className == 'mO')?'mC': 'mO';
  return false;
}
function changeAlt(id)
{
  id.alt = (id.alt == 'Expand Menu')?'Collapse Menu' : 'Expand Menu';
  id.title = (id.title == 'Expand Menu')?'Collapse Menu' : 'Expand Menu';
}
</script>
<!--Left Nav Starts -->


	<ul>			  
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/index.htm">
		                  Course Home  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/syllabus/index.htm">
		                  Syllabus  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/calendar/index.htm">
		                  Calendar  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/readings/index.htm">
		                  Readings  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="selected">
			   			<a href="../../../contents/video-lectures/index.htm">
		                  Video Lectures  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/recitations/index.htm">
		                  Recitations  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/assignments/index.htm">
		                  Assignments  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/exams/index.htm">
		                  Exams  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    
		    
         	
	<!--second tal block close-->  
	
</ul>


<!--Left Nav Ends -->





           			</nav>
           			<main id="course_inner_media" aria-labelledby="media_title">
      					 
        <div class="" id="parent-fieldname-text">
            
            
        </div>
    
      					     
    
    



<script type="text/javascript">var caption_embed_1 ={'English - US': '/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-13-sums-and-asymptotics/X9eErxRjQEI.srt'}</script>     
     <div id="media-embed">
         <div class="attention_message" id="embed_1">
<p>Flash and JavaScript are required for this feature.</p>
<p>Download the video from <a href="http://itunes.apple.com/us/itunes-u/lecture-13-sums-asymptotics/id503873536?i=110644972">iTunes U</a> or the <a href="http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec13_300k.mp4">Internet Archive</a>.</p>
</div>
     </div>
    
     <script type="text/javascript">ocw_embed_chapter_media('embed_1', 'https://www.youtube.com/v/X9eErxRjQEI', 'youtube', '/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-13-sums-and-asymptotics', 'https://img.youtube.com/vi/X9eErxRjQEI/0.jpg',0,0, 'https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-13-sums-and-asymptotics/X9eErxRjQEI.srt')</script>
	 
	 
	 		
	 		<div id="transcript1"></div>
				 <script type="text/javascript">createThreePlayParams(2, 781498, "embed_1", 0, 0)</script>
			 
	

     <div id="media_resource_next_prev_nav" style="margin-top: 1em;">
        <p>
        
            <a href="../../../contents/video-lectures/lecture-12-sums/index.htm">
                <img src="../../../common/images/btn_previous_resource.png" style="margin: 0 30px 0 50px;" alt="Previous track" title="Previous track"></a>
     	
     	
        
            <a href="../../../contents/video-lectures/lecture-14-divide-and-conquer-recurrences/index.htm"> 
                <img src="../../../common/images/btn_next_resource.png" alt="Next track" title="Next track"></a>
       
       </p>
     </div>
 


<script type="text/javascript">
		window.onload=function(){
		init();
		
		}
		var tabLinks = new Array();
		var contentDivs = new Array();
		function init() {
		  // Grab the tab links and content divs from the page
		  var tabListItems = document.getElementById('tabs').childNodes;
		  for ( var i = 0; i < tabListItems.length; i++ ) {
			if ( tabListItems[i].nodeName == "LI" ) {
			  var tabLink = getFirstChildWithTagName( tabListItems[i], 'A' );
			  var id = getHash( tabLink.getAttribute('href') );
			  tabLinks[id] = tabLink;
			  contentDivs[id] = document.getElementById( id );
			}
		  }
		  // Assign onclick events to the tab links, and
		  // highlight the first tab
		  var i = 0;
		  for ( var id in tabLinks ) {
			tabLinks[id].onclick = showTab;
			tabLinks[id].onfocus = function() { this.blur() };
			if ( i == 0 ) tabLinks[id].className = 'selected';
			i++;
		  }
		  // Hide all content divs except the first
		  var i = 0;
		  for ( var id in contentDivs ) {
			if ( i != 0 ) contentDivs[id].className = 'tabContent hide';
			i++;
		  }
		}
		function showTab() {
		  var selectedId = getHash( this.getAttribute('href') );
		  // Highlight the selected tab, and dim all others.
		  // Also show the selected content div, and hide all others.
		  for ( var id in contentDivs ) {
			if ( id == selectedId ) {
			  tabLinks[id].className = 'selected';
			  contentDivs[id].className = 'tabContent';
			} else {
			  tabLinks[id].className = '';
			  contentDivs[id].className = 'tabContent hide';
			}
		  }
		  // Stop the browser following the link
		  return false;
		}
		function getFirstChildWithTagName( element, tagName ) {
		  for ( var i = 0; i < element.childNodes.length; i++ ) {
			if ( element.childNodes[i].nodeName == tagName ) return element.childNodes[i];
		  }
		}
		function getHash( url ) {
		  var hashPos = url.lastIndexOf ( '#' );
		  return url.substring( hashPos + 1 );
		}
 </script>	
 

  <div id="media_tabs">
     
        <ul id="tabs">
            <li class="first">
                <a href="#vid_about" class="selected">About this Video</a>
            </li>
            <li class="">
                <a href="#vid_index" class="">Playlist</a>
            </li>
            <li class="">
                <a href="#vid_playlist" class="">Transcript</a>
            </li>
            <li class="">
                <a href="#vid_related" class="">Download this Video</a>
            </li>
        </ul>
   
        <div id="vid_about" itemprop="description" class="tabContent">
<p><strong>Description:</strong> Analysis of sums, formulation of asymptotic bounds using various techniques, and introduction to asymptotic notation.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
</div>
        <div id="vid_index" itemprop="description" class="tabContent hide">
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-1-introduction-and-proofs/index.htm">
<img src="https://img.youtube.com/vi/L3LMbpZIKhQ/default.jpg" title="Lecture 1: Introduction and Proofs" alt="Lecture 1: Introduction and Proofs">
<p>Lecture 1: Introduction and...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-2-induction/index.htm">
<img src="https://img.youtube.com/vi/z8HKWUWS-lA/default.jpg" title="Lecture 2: Induction" alt="Lecture 2: Induction">
<p>Lecture 2: Induction</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-3-strong-induction/index.htm">
<img src="https://img.youtube.com/vi/NuGDkmwEObM/default.jpg" title="Lecture 3: Strong Induction" alt="Lecture 3: Strong Induction">
<p>Lecture 3: Strong Induction</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-4-number-theory-i/index.htm">
<img src="https://img.youtube.com/vi/NuY7szYSXSw/default.jpg" title="Lecture 4: Number Theory I" alt="Lecture 4: Number Theory I">
<p>Lecture 4: Number Theory I</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-5-number-theory-ii/index.htm">
<img src="https://img.youtube.com/vi/XX7ePR21Ook/default.jpg" title="Lecture 5: Number Theory II" alt="Lecture 5: Number Theory II">
<p>Lecture 5: Number Theory II</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-6-graph-theory-and-coloring/index.htm">
<img src="https://img.youtube.com/vi/h9wxtqoa1jY/default.jpg" title="Lecture 6: Graph Theory and Coloring" alt="Lecture 6: Graph Theory and Coloring">
<p>Lecture 6: Graph Theory and...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-7-matching-problems/index.htm">
<img src="https://img.youtube.com/vi/5RSMLgy06Ew/default.jpg" title="Lecture 7: Matching Problems" alt="Lecture 7: Matching Problems">
<p>Lecture 7: Matching Problems</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-8-graph-theory-ii-minimum-spanning-trees/index.htm">
<img src="https://img.youtube.com/vi/GJpt_3ie4WU/default.jpg" title="Lecture 8: Graph Theory II: Minimum Spanning Trees" alt="Lecture 8: Graph Theory II: Minimum Spanning Trees">
<p>Lecture 8: Graph Theory II:...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-9-communication-networks/index.htm">
<img src="https://img.youtube.com/vi/bTyxpoi2dmM/default.jpg" title="Lecture 9: Communication Networks" alt="Lecture 9: Communication Networks">
<p>Lecture 9: Communication Ne...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-10-graph-theory-iii/index.htm">
<img src="https://img.youtube.com/vi/DOIp5D7VMS4/default.jpg" title="Lecture 10: Graph Theory III" alt="Lecture 10: Graph Theory III">
<p>Lecture 10: Graph Theory III</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-11-relations-partial-orders-and-scheduling/index.htm">
<img src="https://img.youtube.com/vi/1nScXLQAQ9A/default.jpg" title="Lecture 11: Relations, Partial Orders, and Scheduling" alt="Lecture 11: Relations, Partial Orders, and Scheduling">
<p>Lecture 11: Relations, Part...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-12-sums/index.htm">
<img src="https://img.youtube.com/vi/fAeShezAGLE/default.jpg" title="Lecture 12: Sums" alt="Lecture 12: Sums">
<p>Lecture 12: Sums</p></a>
</div>
<div class="related-media-thumbnail-nolink">
<div class="now-playing-resource">Now Playing</div>
<img src="https://img.youtube.com/vi/X9eErxRjQEI/default.jpg" title="Lecture 13: Sums and Asymptotics" alt="Lecture 13: Sums and Asymptotics">
<p>Lecture 13: Sums and Asympt...</p>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-14-divide-and-conquer-recurrences/index.htm">
<img src="https://img.youtube.com/vi/Kqf0uO0oV6s/default.jpg" title="Lecture 14: Divide and Conquer Recurrences" alt="Lecture 14: Divide and Conquer Recurrences">
<p>Lecture 14: Divide and Conq...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-15-linear-recurrences/index.htm">
<img src="https://img.youtube.com/vi/TWBB-JlmYUc/default.jpg" title="Lecture 15: Linear Recurrences" alt="Lecture 15: Linear Recurrences">
<p>Lecture 15: Linear Recurrences</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-16-counting-rules-i/index.htm">
<img src="https://img.youtube.com/vi/pNt5Ll6hGqo/default.jpg" title="Lecture 16: Counting Rules I" alt="Lecture 16: Counting Rules I">
<p>Lecture 16: Counting Rules I</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-17-counting-rules-ii/index.htm">
<img src="https://img.youtube.com/vi/09yIb3VHhMI/default.jpg" title="Lecture 17: Counting Rules II" alt="Lecture 17: Counting Rules II">
<p>Lecture 17: Counting Rules II</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-18-probability-introduction/index.htm">
<img src="https://img.youtube.com/vi/SmFwFdESMHI/default.jpg" title="Lecture 18: Probability Introduction" alt="Lecture 18: Probability Introduction">
<p>Lecture 18: Probability Int...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-19-conditional-probability/index.htm">
<img src="https://img.youtube.com/vi/E6FbvM-FGZ8/default.jpg" title="Lecture 19: Conditional Probability" alt="Lecture 19: Conditional Probability">
<p>Lecture 19: Conditional Pro...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-20-independence/index.htm">
<img src="https://img.youtube.com/vi/l1BCv3qqW4A/default.jpg" title="Lecture 20: Independence" alt="Lecture 20: Independence">
<p>Lecture 20: Independence</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-21-random-variables/index.htm">
<img src="https://img.youtube.com/vi/MOfhhFaQdjw/default.jpg" title="Lecture 21: Random Variables" alt="Lecture 21: Random Variables">
<p>Lecture 21: Random Variables</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-22-expectation-i/index.htm">
<img src="https://img.youtube.com/vi/gGlMSe7uEkA/default.jpg" title="Lecture 22: Expectation I" alt="Lecture 22: Expectation I">
<p>Lecture 22: Expectation I</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-23-expectation-ii/index.htm">
<img src="https://img.youtube.com/vi/oI9fMUqgfxY/default.jpg" title="Lecture 23: Expectation II" alt="Lecture 23: Expectation II">
<p>Lecture 23: Expectation II</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-24-large-deviations/index.htm">
<img src="https://img.youtube.com/vi/q4mwO2qS2z4/default.jpg" title="Lecture 24: Large Deviations" alt="Lecture 24: Large Deviations">
<p>Lecture 24: Large Deviations</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-25-random-walks/index.htm">
<img src="https://img.youtube.com/vi/56iFMY8QW2k/default.jpg" title="Lecture 25: Random Walks" alt="Lecture 25: Random Walks">
<p>Lecture 25: Random Walks</p></a>
</div>
</div>
        <div id="vid_playlist" itemprop="description" class="tabContent hide">
<ul><li><a class="transcript-link" title="Open in a new window." alt="Open in a new window." style="text-decoration: none; font-size: 1.0em;" target="_blank" text-decoration: none font-size: href="../../../contents/video-lectures/lecture-13-sums-and-asymptotics/X9eErxRjQEI.pdf"> Download English-US transcript (PDF)</a></li></ul>
<p><span m="360">The</span> <span m="490">following</span> <span m="940">content</span> <span m="1530">is</span> <span m="1650">provided</span> <span m="2100">under</span> <span m="2350">a</span> <span m="2400">Creative</span> <span m="2800">Commons</span> <span m="3220">license.</span> <span m="4340">Your</span> <span m="4510">support</span> <span m="5050">will</span> <span m="5200">help</span> <span m="5430">MIT</span> <span m="5870">OpenCourseWare</span> <span m="6660">continue</span> <span m="7170">to</span> <span m="7290">offer</span> <span m="7690">high</span> <span m="7910">quality</span> <span m="8420">educational</span> <span m="9060">resources</span> <span m="9640">for</span> <span m="9820">free.</span> <span m="11020">To</span> <span m="11040">make</span> <span m="11250">a</span> <span m="11290">donation</span> <span m="11980">or</span> <span m="12230">view</span> <span m="12680">additional</span> <span m="13120">materials</span> <span m="13640">from</span> <span m="13830">hundreds</span> <span m="14240">of</span> <span m="14360">MIT</span> <span m="14760">courses,</span> <span m="15840">visit</span> <span m="16100">MIT</span> <span m="16510">OpenCourseWare</span> <span m="17560">at</span> <span m="17750">ocw.mit.edu.</span> </p>
<p><span m="22790">PROFESSOR: OK,</span> <span m="23840">let's</span> <span m="23970">get</span> <span m="24130">started.</span> <span m="26120">Last</span> <span m="26430">time</span> <span m="27320">we</span> <span m="27640">looked</span> <span m="27900">at</span> <span m="27980">a</span> <span m="28020">variety</span> <span m="28540">of</span> <span m="28650">ways</span> <span m="29160">to</span> <span m="29500">analyze</span> <span m="30050">sums</span> <span m="30560">and</span> <span m="30650">to</span> <span m="30710">find</span> <span m="31110">closed-form</span> <span m="31920">expressions</span> <span m="32520">for</span> <span m="32640">the</span> <span m="32720">sums.</span> <span m="33900">And</span> <span m="34080">today</span> <span m="34380">we're</span> <span m="34480">going</span> <span m="34600">to</span> <span m="34710">use</span> <span m="35050">those</span> <span m="35370">methods</span> <span m="35830">to</span> <span m="35940">solve</span> <span m="37440">a</span> <span m="37530">variety</span> <span m="37960">of</span> <span m="38190">famous</span> <span m="38640">problems,</span> <span m="39390">one</span> <span m="39620">of</span> <span m="39700">them</span> <span m="39910">involving</span> <span m="41560">center</span> <span m="41920">of</span> <span m="42000">mass</span> <span m="42480">and</span> <span m="42890">stacking</span> <span m="43340">blocks.</span> </p>
<p><span m="44370">We</span> <span m="44620">saw</span> <span m="45220">a</span> <span m="45290">version</span> <span m="45690">of</span> <span m="45760">that</span> <span m="45990">yesterday</span> <span m="46810">in</span> <span m="46900">recitation,</span> <span m="48120">where</span> <span m="48640">you</span> <span m="48770">solved</span> <span m="49130">the</span> <span m="49220">problem</span> <span m="49500">of</span> <span m="49590">figuring</span> <span m="49820">out</span> <span m="50000">how</span> <span m="50290">many</span> <span m="51500">L's</span> <span m="51920">you</span> <span m="52050">need</span> <span m="52900">to</span> <span m="53290">make</span> <span m="53760">something</span> <span m="54010">like</span> <span m="54200">this</span> <span m="54430">balance</span> <span m="55150">perfectly.</span> <span m="56870">Now</span> <span m="57110">in</span> <span m="57200">this</span> <span m="57400">case</span> <span m="58570">I</span> <span m="58720">didn't</span> <span m="59040">actually</span> <span m="59480">get</span> <span m="59610">an</span> <span m="59690">integer</span> <span m="60010">number</span> <span m="60250">of</span> <span m="60430">L's--</span> <span m="60780">which</span> <span m="60980">is</span> <span m="61210">possible--</span> <span m="61610">because</span> <span m="61760">I</span> <span m="61820">didn't</span> <span m="62030">quite</span> <span m="62520">get</span> <span m="62650">it</span> <span m="62750">stacked</span> <span m="63140">right</span> <span m="63300">at</span> <span m="63380">the</span> <span m="63440">bottom.</span> <span m="64230">So</span> <span m="64390">I</span> <span m="64430">got</span> <span m="64670">10</span> <span m="64959">and</span> <span m="65250">1/2</span> <span m="65570">L's</span> <span m="65940">here.</span> <span m="66150">There's</span> <span m="66340">an</span> <span m="66420">extra</span> <span m="67310">vertical</span> <span m="67850">here.</span> </p>
<p><span m="68660">Now</span> <span m="68890">in</span> <span m="68980">theory</span> <span m="69710">there</span> <span m="69990">are</span> <span m="70950">two</span> <span m="71690">integer</span> <span m="72070">ways</span> <span m="72440">of</span> <span m="72540">doing</span> <span m="72800">it.</span> <span m="72990">And</span> <span m="73090">you</span> <span m="73190">proved</span> <span m="73420">that</span> <span m="73900">yesterday</span> <span m="74330">in</span> <span m="74410">recitation.</span> <span m="75430">So</span> <span m="75630">in</span> <span m="75710">theory,</span> <span m="76050">10</span> <span m="76400">and</span> <span m="76640">11</span> <span m="77090">should</span> <span m="77270">work,</span> <span m="78060">but</span> <span m="78230">in</span> <span m="78350">practice</span> <span m="79080">the</span> <span m="79170">blocks</span> <span m="79560">aren't</span> <span m="79770">perfect.</span> <span m="80170">I</span> <span m="80250">didn't</span> <span m="80510">get</span> <span m="80660">them</span> <span m="80800">perfectly</span> <span m="81170">lined</span> <span m="81500">up.</span> <span m="82000">And</span> <span m="82160">so</span> <span m="82310">10</span> <span m="82820">wouldn't</span> <span m="83100">quite</span> <span m="83490">work,</span> <span m="83820">and</span> <span m="83910">11</span> <span m="84350">tended</span> <span m="84640">to</span> <span m="84710">pitch</span> <span m="85020">it</span> <span m="85460">the</span> <span m="85580">other</span> <span m="85730">way.</span> <span m="86530">And</span> <span m="86690">of</span> <span m="86770">course</span> <span m="87170">in</span> <span m="87280">theory,</span> <span m="87750">as</span> <span m="87890">long</span> <span m="88190">as</span> <span m="88280">the</span> <span m="88370">center</span> <span m="88710">of</span> <span m="88820">mass</span> <span m="89890">is</span> <span m="90180">just</span> <span m="90530">anywhere</span> <span m="91120">over</span> <span m="91330">this</span> <span m="91560">block,</span> <span m="92560">it's</span> <span m="92800">going</span> <span m="92930">to</span> <span m="93000">be</span> <span m="93100">stable.</span> <span m="94080">In</span> <span m="94410">practice</span> <span m="95060">there's</span> <span m="95380">air</span> <span m="95610">moving</span> <span m="95880">around,</span> <span m="96440">and</span> <span m="96590">bouncing</span> <span m="97140">and</span> <span m="97240">stuff.</span> <span m="97550">And</span> <span m="97650">so</span> <span m="97740">if</span> <span m="97830">you're</span> <span m="97940">close</span> <span m="98240">to</span> <span m="98320">the</span> <span m="98470">edge,</span> <span m="98830">it's</span> <span m="99220">going</span> <span m="99510">over.</span> </p>
<p><span m="100900">OK.</span> <span m="101420">So</span> <span m="101570">now</span> <span m="101830">if</span> <span m="102010">you</span> <span m="102100">want</span> <span m="102260">to</span> <span m="102320">build</span> <span m="102510">one</span> <span m="102630">of</span> <span m="102690">these</span> <span m="102880">yourself,</span> <span m="103410">it's</span> <span m="103570">not</span> <span m="103830">hard</span> <span m="104050">to</span> <span m="104130">do.</span> <span m="106110">You</span> <span m="106130">just</span> <span m="106330">start</span> <span m="106630">with</span> <span m="106740">an</span> <span m="106830">extra</span> <span m="107070">block</span> <span m="107400">here,</span> <span m="108560">all</span> <span m="108730">right,</span> <span m="108960">and</span> <span m="109050">then</span> <span m="109160">you</span> <span m="109240">build</span> <span m="109570">up.</span> <span m="109780">And</span> <span m="109820">that's</span> <span m="110090">easy.</span> <span m="110380">And</span> <span m="110480">then</span> <span m="110600">you</span> <span m="110690">notice</span> <span m="111200">when</span> <span m="111390">it's</span> <span m="111610">about</span> <span m="111960">right.</span> <span m="113100">And</span> <span m="113200">you</span> <span m="113270">can</span> <span m="113410">just start</span> <span m="113840">wiggling</span> <span m="114040">this</span> <span m="114210">block</span> <span m="114550">out.</span> <span m="114870">And</span> <span m="114990">then</span> <span m="115580">it</span> <span m="115650">looks</span> <span m="115840">pretty</span> <span m="116150">cool</span> <span m="116610">at</span> <span m="116750">that</span> <span m="116950">point.</span> </p>
<p><span m="117850">Actually</span> <span m="118230">it</span> <span m="118260">was</span> <span m="118370">a</span> <span m="118420">student</span> <span m="118760">in</span> <span m="118820">the</span> <span m="118880">class</span> <span m="119250">showed</span> <span m="119520">me</span> <span m="119660">this</span> <span m="121060">a</span> <span m="121140">couple</span> <span m="121390">of</span> <span m="121450">years</span> <span m="121650">ago,</span> <span m="122350">because</span> <span m="122510">we</span> <span m="122590">were</span> <span m="122660">doing</span> <span m="122920">the</span> <span m="122990">blocks.</span> <span m="123390">The</span> <span m="123470">next</span> <span m="123660">thing</span> <span m="123780">we're</span> <span m="123840">going</span> <span m="123960">to</span> <span m="124020">do--</span> <span m="124100">we</span> <span m="124200">were</span> <span m="124320">doing</span> <span m="124570">that--</span> <span m="124780">and he</span> <span m="124850">said,</span> <span m="125170">hey,</span> <span m="125360">there's</span> <span m="125450">this</span> <span m="125640">thing.</span> <span m="125910">And</span> <span m="126230">yeah,</span> <span m="126410">pretty</span> <span m="126600">cool.</span> </p>
<p><span m="128150">So</span> <span m="128310">today</span> <span m="128610">we're</span> <span m="128740">going</span> <span m="128880">to</span> <span m="128940">look</span> <span m="129120">at</span> <span m="129190">a</span> <span m="129240">different</span> <span m="129680">problem,</span> <span m="131039">and</span> <span m="131230">that's</span> <span m="131430">the</span> <span m="131520">problem</span> <span m="132530">of</span> <span m="133020">getting</span> <span m="133890">the</span> <span m="134010">blocks</span> <span m="134540">stacked</span> <span m="134970">up</span> <span m="135150">so</span> <span m="135310">they</span> <span m="135430">go</span> <span m="135660">out</span> <span m="136000">over</span> <span m="136300">the</span> <span m="136450">edge</span> <span m="136640">of</span> <span m="136710">the</span> <span m="136780">table--</span> <span m="137760">if</span> <span m="137990">that's</span> <span m="138270">doable.</span> <span m="139950">So</span> <span m="140280">the</span> <span m="140380">goal</span> <span m="144510">is</span> <span m="144850">to</span> <span m="146400">stack</span> <span m="146920">the</span> <span m="146990">blocks</span> <span m="147540">up</span> <span m="150190">like</span> <span m="150480">this.</span> <span m="150840">We</span> <span m="150930">go</span> <span m="151050">farther</span> <span m="151430">and</span> <span m="151540">farther</span> <span m="151950">out.</span> <span m="152920">And</span> <span m="153140">the</span> <span m="153210">question</span> <span m="153760">we're</span> <span m="153850">going</span> <span m="153970">to</span> <span m="154030">look</span> <span m="154310">at</span> <span m="155410">is</span> <span m="155745">is</span> <span m="156080">it</span> <span m="156220">possible--</span> <span m="157820">whoa.</span> <span m="159500">Almost.</span> <span m="159910">Is</span> <span m="160020">it</span> <span m="160120">possible</span> <span m="160580">to</span> <span m="160660">get</span> <span m="160810">a</span> <span m="160850">block</span> <span m="161930">all</span> <span m="162320">the</span> <span m="162400">way</span> <span m="162630">off</span> <span m="162870">the</span> <span m="162970">table?</span> <span m="164450">That's</span> <span m="164610">the</span> <span m="164680">question we're</span> <span m="165030">going</span> <span m="165150">to</span> <span m="165220">look</span> <span m="165460">at.</span> </p>
<p><span m="165700">And here</span> <span m="165920">I</span> <span m="166030">got</span> <span m="166990">close.</span> <span m="167440">And</span> <span m="167530">you</span> <span m="167590">can</span> <span m="167730">see</span> <span m="168030">it's</span> <span m="168220">getting</span> <span m="168430">a</span> <span m="168470">little</span> <span m="168700">tipsy.</span> <span m="170450">It's</span> <span m="170580">not</span> <span m="170730">going</span> <span m="170860">to</span> <span m="170920">go.</span> <span m="171090">It's</span> <span m="171230">going</span> <span m="171350">to</span> <span m="171410">fall</span> <span m="171690">over.</span> <span m="173450">So</span> <span m="173510">I</span> <span m="173590">didn't</span> <span m="173820">do</span> <span m="174040">it</span> <span m="174160">this</span> <span m="174380">time.</span> </p>
<p><span m="176030">You're</span> <span m="176230">only</span> <span m="176500">allowed</span> <span m="176890">one</span> <span m="177370">block</span> <span m="177970">per</span> <span m="178130">level.</span> <span m="179360">So</span> <span m="179990">you</span> <span m="180080">can't</span> <span m="180400">do</span> <span m="182430">anything</span> <span m="182790">like</span> <span m="183470">this.</span> <span m="184110">That's</span> <span m="184410">not</span> <span m="184630">allowed.</span> <span m="185590">You</span> <span m="185740">can</span> <span m="186070">go</span> <span m="186220">backwards,</span> <span m="188330">if</span> <span m="188420">that</span> <span m="188630">helps</span> <span m="188940">you.</span> <span m="190680">Well</span> <span m="191910">it's</span> <span m="192020">got</span> <span m="192160">to</span> <span m="192230">be</span> <span m="192310">stable</span> <span m="192700">on</span> <span m="192830">both</span> <span m="193190">sides.</span> <span m="193570">But</span> <span m="193680">you</span> <span m="193760">can</span> <span m="194040">go</span> <span m="194190">back</span> <span m="194500">and</span> <span m="194620">forth.</span> </p>
<p><span m="196220">So</span> <span m="196380">the</span> <span m="196450">question</span> <span m="196840">is,</span> <span m="197620">can</span> <span m="197770">you</span> <span m="197850">get</span> <span m="198000">out</span> <span m="198350">one</span> <span m="198610">block</span> <span m="198970">totally</span> <span m="199370">out</span> <span m="199690">over</span> <span m="199890">the</span> <span m="199980">table?</span> <span m="201360">How</span> <span m="201530">many</span> <span m="201710">people</span> <span m="202010">think</span> <span m="202260">it</span> <span m="202400">can't</span> <span m="202890">be</span> <span m="203000">done?</span> <span m="205560">All right.</span> <span m="205660">A</span> <span m="205760">few</span> <span m="205960">doubters.</span> <span m="206810">More</span> <span m="207040">doubters.</span> </p>
<p><span m="207830">How</span> <span m="208010">many</span> <span m="208180">think</span> <span m="208440">it</span> <span m="208550">can</span> <span m="208830">be</span> <span m="208930">done?</span> <span m="210180">Oh,</span> <span m="210460">a</span> <span m="210490">lot</span> <span m="210780">of</span> <span m="210840">people</span> <span m="211150">think</span> <span m="211340">it</span> <span m="211420">can</span> <span m="211540">be</span> <span m="211640">done.</span> <span m="211930">OK.</span> <span m="212750">Well</span> <span m="213360">all</span> <span m="213710">right.</span> </p>
<p><span m="215300">How</span> <span m="215520">many</span> <span m="215740">people</span> <span m="216080">think</span> <span m="216360">you</span> <span m="216460">can</span> <span m="216640">get</span> <span m="216800">a</span> <span m="216840">block</span> <span m="218540">two</span> <span m="218860">blocks</span> <span m="219330">out</span> <span m="219540">over</span> <span m="219720">the</span> <span m="219820">table?</span> <span m="222080">A</span> <span m="222130">few</span> <span m="222440">of</span> <span m="222540">you.</span> <span m="223940">How</span> <span m="224070">many</span> <span m="224250">think</span> <span m="224480">you</span> <span m="224570">can</span> <span m="224750">get it</span> <span m="224940">three</span> <span m="225440">blocks</span> <span m="225870">out?</span> <span m="228230">You're</span> <span m="228380">believers.</span> <span m="229020">OK.</span> <span m="230430">How</span> <span m="230620">many</span> <span m="230820">people</span> <span m="231100">think</span> <span m="231260">I</span> <span m="231320">can</span> <span m="231470">reach</span> <span m="231680">that</span> <span m="231890">wall?</span> <span m="233840">Well</span> <span m="233950">I</span> <span m="233990">got</span> <span m="234110">to</span> <span m="234180">tell</span> <span m="234270">you</span> <span m="234350">I</span> <span m="234680">can't</span> <span m="235020">reach</span> <span m="235250">the</span> <span m="235330">wall</span> <span m="235760">because--</span> </p>
<p><span m="236540">AUDIENCE: The ceiling gets in the way.</span> </p>
<p><span m="237430">PROFESSOR: Yeah.</span> <span m="239020">That's</span> <span m="239340">going</span> <span m="239460">to</span> <span m="239520">be</span> <span m="239590">hard</span> <span m="239860">to</span> <span m="239930">do.</span> <span m="240900">But</span> <span m="241010">this</span> <span m="241190">is</span> <span m="241280">what</span> <span m="241430">we're</span> <span m="241990">going</span> <span m="242110">to</span> <span m="242170">try</span> <span m="242320">to</span> <span m="242410">figure</span> <span m="242750">out.</span> <span m="243560">And</span> <span m="243730">we're</span> <span m="243800">going</span> <span m="243930">to</span> <span m="244000">do</span> <span m="244200">it</span> <span m="244280">with</span> <span m="244390">a</span> <span m="244430">little</span> <span m="244630">competition.</span> <span m="246280">So</span> <span m="246410">we're</span> <span m="246530">going</span> <span m="246650">to</span> <span m="246730">have</span> <span m="247010">a</span> <span m="247070">team</span> <span m="247370">of</span> <span m="247500">you</span> <span m="248600">go</span> <span m="248770">up</span> <span m="248900">against</span> <span m="249690">the</span> <span m="249800">TAs.</span> <span m="252050">And</span> <span m="252570">the</span> <span m="252910">winner</span> <span m="254310">gets</span> <span m="254590">candy</span> <span m="255410">to</span> <span m="255540">share.</span> <span m="257079">The</span> <span m="257230">losing</span> <span m="257680">team</span> <span m="258600">gets</span> <span m="258870">the</span> <span m="259040">infamous</span> <span m="259480">6042</span> <span m="260350">Nerd</span> <span m="260760">Pride</span> <span m="261170">pocket</span> <span m="261519">protectors.</span> <span m="263030">Very</span> <span m="263350">nice.</span> <span m="263940">Shiny</span> <span m="264330">plastic.</span> <span m="264990">Has</span> <span m="265200">the</span> <span m="265290">MIT</span> <span m="265760">logo</span> <span m="266400">on</span> <span m="266610">them.</span> <span m="267100">Very</span> <span m="267410">handy.</span> </p>
<p><span m="268320">All</span> <span m="268600">right.</span> <span m="268810">So</span> <span m="269090">I</span> <span m="269210">need</span> <span m="270140">four</span> <span m="270530">volunteers.</span> <span m="271730">So</span> <span m="271950">come</span> <span m="272100">on</span> <span m="272280">down</span> <span m="272650">if</span> <span m="272750">you'd</span> <span m="272760">like</span> <span m="272920">to</span> <span m="273000">volunteer.</span> <span m="273710">You</span> <span m="273840">can</span> <span m="273990">win</span> <span m="274230">some</span> <span m="274420">candy</span> <span m="274750">for</span> <span m="274870">the</span> <span m="274950">class--</span> <span m="275400">or</span> <span m="275460">at</span> <span m="275540">least</span> <span m="276000">win</span> <span m="276330">yourself</span> <span m="276810">a</span> <span m="276870">pocket</span> <span m="277230">protector.</span> <span m="278810">And</span> <span m="279130">we'll</span> <span m="279240">put</span> <span m="279480">the</span> <span m="279550">class</span> <span m="279960">over</span> <span m="280150">there.</span> </p>
<p><span m="282020">And</span> <span m="282470">we</span> <span m="282530">get</span> <span m="282650">the</span> <span m="282710">TAs</span> <span m="283150">to</span> <span m="283230">come</span> <span m="283450">down.</span> <span m="285610">So</span> <span m="285780">come</span> <span m="285940">on</span> <span m="286090">down.</span> <span m="286460">I</span> <span m="286570">got</span> <span m="286680">a</span> <span m="286710">couple</span> <span m="286960">of</span> <span m="287060">people.</span> <span m="287220">Who</span> <span m="287320">else</span> <span m="287540">wants</span> <span m="287710">to</span> <span m="287820">join</span> <span m="288150">them?</span> <span m="289170">All right.</span> <span m="289420">Come</span> <span m="289580">on</span> <span m="289760">down.</span> </p>
<p><span m="290910">We've</span> <span m="291010">got</span> <span m="291130">a</span> <span m="291170">fourth</span> <span m="291570">volunteer</span> <span m="293650">out</span> <span m="293900">there?</span> <span m="297322">All right.</span> <span m="297770">We</span> <span m="297870">got</span> <span m="298040">some</span> <span m="298160">people</span> <span m="298440">coming</span> <span m="298680">down.</span> <span m="298910">All right,</span> <span m="299060">come</span> <span m="299250">on</span> <span m="299430">over.</span> <span m="300620">All right.</span> <span m="300930">You</span> <span m="301010">got</span> <span m="301090">all</span> <span m="301350">the</span> <span m="301420">blocks</span> <span m="301810">you</span> <span m="301890">want.</span> <span m="302270">We</span> <span m="302360">could</span> <span m="302470">even</span> <span m="302640">take</span> <span m="302880">this</span> <span m="303080">thing</span> <span m="303290">down.</span> <span m="304660">So</span> <span m="304850">here's</span> <span m="305760">your</span> <span m="305880">supply.</span> <span m="308660">All</span> <span m="308830">right.</span> </p>
<p><span m="309490">Now</span> <span m="309840">the</span> <span m="309950">TAs</span> <span m="310400">have</span> <span m="310590">a</span> <span m="310630">little</span> <span m="310870">bit</span> <span m="311000">of</span> <span m="311100">an</span> <span m="311160">advantage,</span> <span m="311810">because</span> <span m="312810">they've</span> <span m="313130">done</span> <span m="313370">this</span> <span m="313540">before,</span> <span m="314170">some</span> <span m="314370">of</span> <span m="314450">them.</span> <span m="315960">So</span> <span m="316160">you</span> <span m="316320">can</span> <span m="316490">give</span> <span m="316650">some</span> <span m="316780">advice</span> <span m="317280">to</span> <span m="317420">your</span> <span m="317680">colleagues</span> <span m="318250">here.</span> <span m="319070">So</span> <span m="319250">you</span> <span m="319320">got</span> <span m="319420">to</span> <span m="319480">get</span> <span m="319610">some</span> <span m="319730">blocks</span> <span m="320100">out</span> <span m="320270">of</span> <span m="320330">the</span> <span m="320400">block</span> <span m="320730">supply.</span> </p>
<p><span m="321460">Maybe</span> <span m="321680">we</span> <span m="321760">should</span> <span m="321920">take</span> <span m="322160">this</span> <span m="322330">down.</span> <span m="322560">It's</span> <span m="322690">about</span> <span m="322900">to</span> <span m="322970">fall</span> <span m="323300">over</span> <span m="323550">anyway.</span> <span m="325180">So</span> <span m="325430">yeah,</span> <span m="325930">how</span> <span m="326030">do</span> <span m="326110">we</span> <span m="326220">get</span> <span m="326400">this</span> <span m="326590">down?</span> <span m="327550">I'm</span> <span m="328030">not</span> <span m="328240">sure.</span> <span m="330576">All</span> <span m="331030">right,</span> <span m="331150">I got this</span> <span m="331460">end.</span> <span m="334476">Whoa!</span> <span m="335380">There</span> <span m="335600">we</span> <span m="335700">go.</span> <span m="335940">Whoa!</span> </p>
<p><span m="336940">[CRASHING BLOCKS]</span> </p>
<p><span m="337940">Oops!</span> <span m="338940">All right.</span> <span m="341910">There we go.</span> </p>
<p><span m="350020">So</span> <span m="350270">whoever</span> <span m="350660">gets</span> <span m="350900">a</span> <span m="350950">block</span> <span m="351340">out</span> <span m="351590">over</span> <span m="351870">the</span> <span m="351950">farthest</span> <span m="352780">wins.</span> <span m="353870">One</span> <span m="354260">block</span> <span m="354580">per</span> <span m="354740">level.</span> <span m="355240">You</span> <span m="355370">can't</span> <span m="355660">do</span> <span m="355880">two</span> <span m="355970">blocks</span> <span m="356300">on</span> <span m="356400">a</span> <span m="356450">level.</span> <span m="357760">Believe</span> <span m="357990">me</span> <span m="358060">it's</span> <span m="358170">a</span> <span m="358240">lot</span> <span m="358530">easier</span> <span m="358940">if</span> <span m="359020">you</span> <span m="359120">can</span> <span m="359250">do</span> <span m="359410">multiple</span> <span m="359800">blocks</span> <span m="360110">per</span> <span m="360260">level.</span> </p>
<p><span m="365080">You</span> <span m="365270">can</span> <span m="365620">go</span> <span m="365790">back</span> <span m="366070">and</span> <span m="366190">forth.</span> <span m="366590">Go</span> <span m="366710">ahead</span> <span m="366880">and</span> <span m="366970">give</span> <span m="367100">them</span> <span m="367260">some</span> <span m="367430">advice.</span> <span m="368680">They</span> <span m="368860">look</span> <span m="369030">like</span> <span m="369200">they</span> <span m="369340">need</span> <span m="369560">it.</span> <span m="371280">They're</span> <span m="371450">only</span> <span m="371650">about</span> <span m="371900">half</span> <span m="372120">a</span> <span m="372170">block</span> <span m="372510">out</span> <span m="372730">so</span> <span m="372900">far.</span> <span m="376590">How are</span> <span m="376840">the</span> <span m="376930">TAs</span> <span m="377420">doing</span> <span m="377730">here?</span> </p>
<p><span m="378591">AUDIENCE: Three steps forward,</span> <span m="379052">two steps back?</span> <span m="379513">Something like that?</span> </p>
<p><span m="380435">[INTERPOSING VOICES]</span> </p>
<p><span m="385780">This is</span> <span m="386280">an interesting idea</span> <span m="386780">I think</span> <span m="387280">is</span> <span m="387748">to go out</span> <span m="388216">as far as</span> <span m="388684">possible</span> <span m="389152">and then just weight it</span> <span m="389620">back.</span> <span m="390556">Does that</span> <span m="391030">change anything?</span> </p>
<p><span m="392858">Like this you're saying?</span> </p>
<p><span m="395800">PROFESSOR: All</span> <span m="395860">right.</span> <span m="395970">The</span> <span m="396050">TAs</span> <span m="396470">are</span> <span m="396520">getting</span> <span m="396770">scientific</span> <span m="397490">here.</span> </p>
<p><span m="398454">AUDIENCE: It need to be at</span> <span m="398936">least half.</span> </p>
<p><span m="400220">PROFESSOR: You</span> <span m="400350">know</span> <span m="400510">it's</span> <span m="400640">not</span> <span m="400880">looking</span> <span m="401150">good</span> <span m="401370">for</span> <span m="401470">you</span> <span m="401590">guys</span> <span m="401850">to</span> <span m="401920">get</span> <span m="402090">the</span> <span m="402160">candy</span> <span m="402550">today.</span> <span m="402850">I</span> <span m="402910">got</span> <span m="403060">to</span> <span m="403140">tell</span> <span m="403400">you.</span> <span m="409560">Ooh.</span> <span m="410035">TAs</span> <span m="410510">are</span> <span m="410570">close.</span> </p>
<p><span m="413570">AUDIENCE: No, that's too</span> <span m="414070">far.</span> <span m="414570">Too far.</span> </p>
<p><span m="415570">Look at the distance</span> <span m="416027">between this and</span> <span m="416484">this.</span> </p>
<p><span m="417398">[INTERPOSING VOICES]</span> </p>
<p><span m="426420">PROFESSOR: All right.</span> <span m="426840">Oh,</span> <span m="427060">you're</span> <span m="427210">about</span> <span m="427420">2/3</span> <span m="427910">of</span> <span m="427970">the</span> <span m="428030">way</span> <span m="428220">over.</span> <span m="428580">We've</span> <span m="428700">got</span> <span m="428870">one</span> <span m="429070">2/3.</span> <span m="431540">You've got</span> <span m="431700">an</span> <span m="431780">inch</span> <span m="431980">to</span> <span m="432080">go.</span> <span m="433250">Yeah.</span> <span m="433390">Maybe</span> <span m="433680">it's</span> <span m="433830">not</span> <span m="434060">doable.</span> <span m="434530">It's</span> <span m="434540">not</span> <span m="434710">the</span> <span m="434780">first</span> <span m="435100">time</span> <span m="435300">we've</span> <span m="435450">done</span> <span m="435670">this</span> <span m="437330">and</span> <span m="437550">the</span> <span m="437620">problem</span> <span m="437970">has</span> <span m="438050">not</span> <span m="438280">been</span> <span m="438741">solvable.</span> </p>
<p><span m="440585">AUDIENCE: The TAs are</span> <span m="441050">doing pretty well.</span> </p>
<p><span m="442280">PROFESSOR: Oh.</span> <span m="442645">TAs</span> <span m="443010">are</span> <span m="443070">close.</span> <span m="443900">they're</span> <span m="444080">definitely</span> <span m="444520">within</span> <span m="444750">an</span> <span m="444870">inch.</span> <span m="447130">Maybe</span> <span m="447270">an</span> <span m="447340">inch</span> <span m="447500">and</span> <span m="447590">a</span> <span m="447640">half.</span> </p>
<p><span m="453986">AUDIENCE: Oh, yeah,</span> <span m="454440">the top one</span> <span m="454620">should</span> <span m="455100">do</span> <span m="455580">[INAUDIBLE].</span> </p>
<p><span m="461350">PROFESSOR: Yeah.</span> <span m="461570">You</span> <span m="461680">can't</span> <span m="461970">hold</span> <span m="462250">it.</span> <span m="462320">It's</span> <span m="462430">got</span> <span m="462620">to</span> <span m="462700">stand</span> <span m="463030">up</span> <span m="463130">by</span> <span m="463270">itself.</span> </p>
<p><span m="467760">Now</span> <span m="467910">the</span> <span m="468000">bottom</span> <span m="468380">block</span> <span m="468800">can go</span> <span m="468940">out</span> <span m="469080">over</span> <span m="469230">the</span> <span m="469350">edge</span> <span m="469570">if</span> <span m="469680">you</span> <span m="469790">want.</span> <span m="473730">Oh,</span> <span m="474230">they're</span> <span m="474340">getting</span> <span m="474610">close.</span> <span m="476500">Oh,</span> <span m="476770">I think</span> <span m="476930">you</span> <span m="477460">might</span> <span m="477900">have</span> <span m="478340">the</span> <span m="478420">top</span> <span m="478700">block</span> <span m="479020">out</span> <span m="479240">over.</span> <span m="479570">Maybe.</span> <span m="483240">How</span> <span m="483525">are</span> <span m="483810">you</span> <span m="483940">guys</span> <span m="484240">doing?</span> <span m="491230">Not</span> <span m="491490">good.</span> <span m="492510">This</span> <span m="493330">is</span> <span m="493870">trouble</span> <span m="494320">for</span> <span m="494430">the--</span> </p>
<p><span m="498150">AUDIENCE: It's out.</span> </p>
<p><span m="498970">PROFESSOR: You've</span> <span m="499110">got</span> <span m="499340">it?</span> </p>
<p><span m="499920">AUDIENCE: Yeah.</span> </p>
<p><span m="500820">PROFESSOR: Ooh. All right.</span> <span m="501050">You've</span> <span m="501150">got</span> <span m="501280">to</span> <span m="501670">work</span> <span m="501860">fast</span> <span m="502220">now if</span> <span m="502350">you're</span> <span m="502470">going</span> <span m="502590">to</span> <span m="502650">beat</span> <span m="502860">that.</span> <span m="504100">TAs</span> <span m="504550">got</span> <span m="504760">it</span> <span m="504840">out</span> <span m="505010">over</span> <span m="505200">the</span> <span m="505350">edge.</span> <span m="505660">You</span> <span m="505720">see</span> <span m="505860">that?</span> <span m="506360">They're out</span> <span m="506630">over</span> <span m="506760">the</span> <span m="506890">edge.</span> <span m="508650">All</span> <span m="508840">right.</span> <span m="508970">I</span> <span m="509010">think</span> <span m="509180">they</span> <span m="509290">did</span> <span m="509490">it.</span> </p>
<p><span m="510484">[INTERPOSING VOICES]</span> </p>
<p><span m="513570">We're</span> <span m="513730">going</span> <span m="513850">to</span> <span m="513909">have</span> <span m="514070">a</span> <span m="514169">hard</span> <span m="514390">time</span> <span m="514650">analyzing</span> <span m="515210">that</span> <span m="515419">one.</span> </p>
<p><span m="519103">AUDIENCE: Uh,</span> <span m="519595">we're--</span> <span m="520579">I think we're</span> <span m="521071">actually there.</span> </p>
<p><span m="522055">PROFESSOR: No.</span> <span m="526490">All</span> <span m="526800">right. Well</span> <span m="526910">try</span> <span m="527170">another</span> <span m="527430">block</span> <span m="527770">or</span> <span m="527830">two,</span> <span m="528040">and</span> <span m="528140">then</span> <span m="528300">we'll</span> <span m="529080">vote.</span> <span m="532080">See?</span> </p>
<p><span m="532580">AUDIENCE: It's on the</span> <span m="533080">edge.</span> <span m="534080">Right</span> <span m="534410">there.</span> </p>
<p><span m="534960">PROFESSOR: All right.</span> </p>
<p><span m="536856">AUDIENCE: Looks like</span> <span m="537330">the Stata Center.</span> </p>
<p><span m="538280">PROFESSOR: Yeah</span> <span m="538710">that's</span> <span m="538980">it.</span> </p>
<p><span m="540390">AUDIENCE: I'm holding it.</span> </p>
<p><span m="541750">PROFESSOR: You're</span> <span m="541930">close.</span> </p>
<p><span m="548410">[INTERPOSING VOICES]</span> </p>
<p><span m="550650">PROFESSOR: You're</span> <span m="550840">not</span> <span m="551080">the</span> <span m="551150">most</span> <span m="551410">efficient,</span> <span m="551830">that's</span> <span m="552010">for</span> <span m="552130">sure.</span> <span m="557740">Yeah</span> <span m="557940">the</span> <span m="558030">TAs</span> <span m="558500">are</span> <span m="558600">already</span> <span m="558820">munching</span> <span m="559180">down</span> <span m="559510">here.</span> </p>
<p><span m="560050">[INTERPOSING VOICES]</span> </p>
<p></p>
<p><span m="569410">AUDIENCE: Copy the TAs'</span> <span m="569910">solution.</span> </p>
<p><span m="570910">PROFESSOR: Now</span> <span m="571120">there's</span> <span m="571430">an</span> <span m="571530">idea.</span> <span m="575220">All</span> <span m="575610">right.</span> <span m="575820">Well</span> <span m="577000">I</span> <span m="577150">think</span> <span m="577340">it's</span> <span m="577450">time</span> <span m="577740">for</span> <span m="577880">a</span> <span m="577930">vote.</span> <span m="579560">But</span> <span m="579710">let</span> <span m="579870">the</span> <span m="579940">class</span> <span m="580310">see</span> <span m="580460">what</span> <span m="580620">you</span> <span m="580690">did.</span> <span m="583290">Yeah</span> <span m="583440">you</span> <span m="583530">got</span> <span m="583650">to</span> <span m="583710">take</span> <span m="583840">your</span> <span m="583950">hands</span> <span m="584290">off.</span> <span m="584560">That's</span> <span m="584740">you know.</span> </p>
<p><span m="585270">AUDIENCE: We can. We're just choosing</span> <span m="585640">not to.</span> </p>
<p><span m="589380">PROFESSOR: Get</span> <span m="589550">some</span> <span m="589760">good</span> <span m="590000">top</span> <span m="590480">balancing</span> <span m="590980">on</span> <span m="591100">there.</span> <span m="594714">I</span> <span m="595196">don't know.</span> <span m="600500">All right.</span> <span m="600740">Can</span> <span m="600970">you let</span> <span m="601100">it</span> <span m="601160">go?</span> <span m="602280">All right.</span> <span m="602470">They</span> <span m="602610">let</span> <span m="602750">it</span> <span m="602810">go.</span> <span m="603070">Let's</span> <span m="603230">see.</span> <span m="603690">Did</span> <span m="603830">you</span> <span m="603900">get</span> <span m="604080">one</span> <span m="604240">out?</span> </p>
<p><span m="605020">AUDIENCE: Yeah.</span> <span m="605477">This one right</span> <span m="605934">here.</span> </p>
<p><span m="606850">PROFESSOR: All</span> <span m="607150">right. We're</span> <span m="607280">going</span> <span m="607400">to</span> <span m="607460">do</span> <span m="607560">a</span> <span m="607640">vote.</span> <span m="608320">Who</span> <span m="608570">thinks</span> <span m="609420">they</span> <span m="609710">got</span> <span m="610240">the</span> <span m="610450">one</span> <span m="610740">farthest</span> <span m="611230">out?</span> <span m="613350">Oh</span> <span m="613610">it's</span> <span m="613740">close.</span> <span m="614790">Ah!</span> <span m="615230">Look at that.</span> <span m="616550">We</span> <span m="616670">need</span> <span m="616840">some</span> <span m="616990">glasses</span> <span m="617470">out</span> <span m="617650">there.</span> <span m="618080">Who</span> <span m="618330">thinks</span> <span m="618980">the</span> <span m="619100">TAs</span> <span m="619570">got it</span> <span m="619830">farther</span> <span m="620190">out?</span> </p>
<p><span m="621186">AUDIENCE: Boo.</span> </p>
<p><span m="622182">PROFESSOR: Oh</span> <span m="622680">man.</span> <span m="623050">All</span> <span m="623420">right.</span> <span m="625820">All</span> <span m="625910">right. So</span> <span m="626200">you</span> <span m="626360">guys.</span> <span m="627270">Here</span> <span m="627440">you</span> <span m="627540">go.</span> <span m="628050">Your</span> <span m="628180">very</span> <span m="628430">nice</span> <span m="628730">pocket</span> <span m="629110">protectors.</span> <span m="629650">A</span> <span m="629710">good</span> <span m="629910">effort.</span> <span m="630865">There</span> <span m="631340">you</span> <span m="631520">are.</span> <span m="631840">Well</span> <span m="632100">done.</span> </p>
<p><span m="633860">And</span> <span m="634210">I'm</span> <span m="634340">afraid</span> <span m="635760">they</span> <span m="635960">get</span> <span m="636120">the</span> <span m="636200">candy</span> <span m="636610">here.</span> <span m="637480">So</span> <span m="637640">you</span> <span m="637760">can</span> <span m="637940">take</span> <span m="638170">these</span> <span m="638400">out</span> <span m="638580">and</span> <span m="638710">share</span> <span m="639020">them</span> <span m="639300">with</span> <span m="639410">the</span> <span m="639490">class.</span> <span m="642660">Oh, leave</span> <span m="642850">it that way.</span> <span m="643100">Leave</span> <span m="643270">it</span> <span m="643340">there.</span> <span m="643500">I</span> <span m="643610">mean,</span> <span m="643870">we're</span> <span m="643940">going</span> <span m="644060">to</span> <span m="644120">analyze</span> <span m="644580">yours,</span> <span m="644970">because</span> <span m="645160">I</span> <span m="645260">can't</span> <span m="645520">begin</span> <span m="645780">to</span> <span m="645850">analyze</span> <span m="646220">that.</span> </p>
<p><span m="650340">All</span> <span m="650640">right. Now</span> <span m="650840">watching</span> <span m="651430">this,</span> <span m="651770">how</span> <span m="651950">many</span> <span m="652160">people</span> <span m="652460">still</span> <span m="652860">think</span> <span m="653160">we</span> <span m="653260">can</span> <span m="653400">get</span> <span m="653520">it</span> <span m="653660">all</span> <span m="653800">the</span> <span m="653870">way</span> <span m="654020">to</span> <span m="654090">the</span> <span m="654170">wall?</span> <span m="656020">Ah,</span> <span m="656210">a</span> <span m="656270">few</span> <span m="656450">people.</span> <span m="656860">In</span> <span m="656960">fact,</span> <span m="657910">if</span> <span m="658220">you</span> <span m="658420">stack</span> <span m="658850">this</span> <span m="659050">high</span> <span m="659380">enough--</span> <span m="659830">and</span> <span m="659950">we're</span> <span m="660010">going</span> <span m="660130">to</span> <span m="660190">analyze</span> <span m="660740">how</span> <span m="661040">high</span> <span m="661270">it</span> <span m="661340">has</span> <span m="661570">to</span> <span m="661670">go--</span> <span m="662250">you</span> <span m="662440">could</span> <span m="662620">reach</span> <span m="662880">the</span> <span m="662970">wall.</span> <span m="663870">Now</span> <span m="664140">in</span> <span m="664230">fact</span> <span m="664520">we're</span> <span m="664580">going</span> <span m="664700">to</span> <span m="664760">find</span> <span m="665390">that</span> <span m="665510">you</span> <span m="665600">probably</span> <span m="665990">got</span> <span m="666160">to</span> <span m="666220">go</span> <span m="666350">out</span> <span m="666500">to</span> <span m="666590">the</span> <span m="666680">stars</span> <span m="667910">to</span> <span m="668060">get</span> <span m="668270">that</span> <span m="668520">high</span> <span m="669010">before</span> <span m="669420">you</span> <span m="669530">can</span> <span m="669690">reach</span> <span m="669910">the</span> <span m="670000">wall.</span> <span m="670490">So</span> <span m="670540">it's</span> <span m="670680">not</span> <span m="671040">too</span> <span m="671240">practical.</span> </p>
<p><span m="673750">Now</span> <span m="674250">what</span> <span m="674390">I</span> <span m="674470">want</span> <span m="674660">to</span> <span m="674720">do</span> <span m="675620">is</span> <span m="675880">analyze</span> <span m="676700">this</span> <span m="676920">strategy.</span> <span m="678200">And</span> <span m="678320">this</span> <span m="678470">sort</span> <span m="678840">of</span> <span m="678910">looks</span> <span m="679080">like</span> <span m="679190">a</span> <span m="679230">nice</span> <span m="679490">design.</span> <span m="679880">They</span> <span m="680020">got</span> <span m="680190">out</span> <span m="680370">there</span> <span m="680500">with</span> <span m="680630">just</span> <span m="680890">five.</span> <span m="681800">And</span> <span m="681970">they</span> <span m="682090">did</span> <span m="682300">a</span> <span m="682600">particular</span> <span m="683830">approach</span> <span m="685070">using</span> <span m="685530">what</span> <span m="685810">we</span> <span m="685950">could</span> <span m="686090">call</span> <span m="686460">as the</span> <span m="686580">greedy</span> <span m="686940">algorithm.</span> <span m="687500">We've</span> <span m="687860">seen</span> <span m="688090">greedy</span> <span m="688380">algorithms</span> <span m="688740">before.</span> <span m="689580">And</span> <span m="689770">they</span> <span m="689920">used</span> <span m="690170">another</span> <span m="690510">greedy</span> <span m="690850">algorithm</span> <span m="691230">here.</span> </p>
<p><span m="692370">Now</span> <span m="693990">what</span> <span m="694280">their</span> <span m="694460">greedy</span> <span m="694820">strategy</span> <span m="695340">did--</span> <span m="695960">and</span> <span m="696080">I'm</span> <span m="696140">going</span> <span m="696260">to</span> <span m="696330">redo</span> <span m="696730">it</span> <span m="696820">here--</span> <span m="698870">is</span> <span m="699910">stack</span> <span m="700310">all</span> <span m="700490">the</span> <span m="700570">blocks</span> <span m="701130">right</span> <span m="701330">at</span> <span m="701410">the</span> <span m="701580">edge.</span> <span m="702770">Line</span> <span m="702960">it</span> <span m="703050">up</span> <span m="703180">here</span> <span m="705010">and,</span> <span m="705180">starting</span> <span m="705590">at</span> <span m="705670">the</span> <span m="705750">top,</span> <span m="706680">push</span> <span m="707020">it</span> <span m="707110">out</span> <span m="707650">as</span> <span m="707870">far</span> <span m="708290">as</span> <span m="708410">you</span> <span m="708530">can</span> <span m="708720">go.</span> <span m="710230">Now</span> <span m="710460">in</span> <span m="710540">this</span> <span m="710720">case,</span> <span m="711260">we</span> <span m="711430">can</span> <span m="711590">get</span> <span m="711750">it</span> <span m="711830">out</span> <span m="711980">about</span> <span m="712220">halfway,</span> <span m="713250">because</span> <span m="713450">the</span> <span m="713510">center</span> <span m="713860">of</span> <span m="713960">mass</span> <span m="714360">of</span> <span m="714450">this</span> <span m="714640">block</span> <span m="715270">is</span> <span m="715450">at</span> <span m="715580">halfway</span> <span m="716050">point.</span> <span m="716920">And</span> <span m="717040">that</span> <span m="717200">better</span> <span m="717420">be</span> <span m="717540">sitting</span> <span m="717910">over</span> <span m="718080">this</span> <span m="718300">block,</span> <span m="718680">or</span> <span m="718790">it's</span> <span m="718950">going</span> <span m="719080">to</span> <span m="719530">tip.</span> <span m="721020">So</span> <span m="721190">the</span> <span m="721270">top</span> <span m="721620">block</span> <span m="721890">goes</span> <span m="722150">out</span> <span m="722350">here.</span> </p>
<p><span m="723860">Now</span> <span m="724090">I'm</span> <span m="724190">going</span> <span m="724310">to</span> <span m="724380">slide</span> <span m="725120">the</span> <span m="725230">top</span> <span m="725700">two</span> <span m="725970">blocks</span> <span m="726400">out</span> <span m="726590">as</span> <span m="726700">far</span> <span m="727030">as</span> <span m="727140">they</span> <span m="727320">go.</span> <span m="731240">All</span> <span m="731710">right. That's</span> <span m="731870">about</span> <span m="732050">the</span> <span m="732120">tip.</span> <span m="733170">Then</span> <span m="733540">I</span> <span m="733660">slide</span> <span m="734130">the</span> <span m="734200">top</span> <span m="734540">three</span> <span m="734890">out</span> <span m="735130">as</span> <span m="735240">far</span> <span m="735550">as</span> <span m="735670">they</span> <span m="735830">go.</span> <span m="738500">Whoa!</span> <span m="739840">Almost.</span> <span m="742600">Oops.</span> <span m="744020">Theory</span> <span m="744300">is</span> <span m="744580">always</span> <span m="744900">easier</span> <span m="745170">than</span> <span m="745310">practice.</span> </p>
<p><span m="747030">And</span> <span m="747300">now</span> <span m="747670">we</span> <span m="747800">do</span> <span m="748780">the</span> <span m="748900">top</span> <span m="749210">four</span> <span m="749480">blocks.</span> <span m="750600">And</span> <span m="751990">they</span> <span m="752150">did</span> <span m="752300">a</span> <span m="752340">better</span> <span m="752540">job</span> <span m="752790">than</span> <span m="752910">I</span> <span m="753020">did.</span> <span m="754120">But</span> <span m="754240">that's</span> <span m="754470">the</span> <span m="754650">algorithm,</span> <span m="755220">OK,</span> <span m="755520">the</span> <span m="755600">greedy</span> <span m="755830">algorithm.</span> <span m="756960">The</span> <span m="757080">top</span> <span m="757480">i</span> <span m="757690">blocks--</span> <span m="758140">when i</span> <span m="758360">goes</span> <span m="758560">1</span> <span m="758800">to</span> <span m="758920">n--</span> <span m="759520">push</span> <span m="759790">it</span> <span m="759870">out</span> <span m="760020">as</span> <span m="760130">far</span> <span m="760370">as</span> <span m="760460">you</span> <span m="760550">can</span> <span m="760710">go.</span> <span m="761750">All</span> <span m="761870">right.</span> <span m="761980">And</span> <span m="762070">we're</span> <span m="762210">going</span> <span m="762340">to</span> <span m="762790">analyze</span> <span m="763400">that</span> <span m="763600">strategy</span> <span m="767840">and</span> <span m="767990">see</span> <span m="768150">how</span> <span m="768310">far</span> <span m="768580">it</span> <span m="768660">goes.</span> </p>
<p><span m="777750">So</span> <span m="777920">we're</span> <span m="778080">given</span> <span m="780640">n</span> <span m="780970">blocks.</span> <span m="782410">And</span> <span m="782590">let's</span> <span m="782770">say</span> <span m="782950">they</span> <span m="783080">all</span> <span m="783220">have</span> <span m="783450">length</span> <span m="783790">1,</span> <span m="784540">to</span> <span m="784650">make</span> <span m="784850">it</span> <span m="784930">easy.</span> <span m="790130">And</span> <span m="790300">we're</span> <span m="790420">going</span> <span m="790560">to</span> <span m="790630">define</span> <span m="792530">r</span> <span m="792810">sub</span> <span m="793195">i</span> <span m="793810">to</span> <span m="793940">be</span> <span m="794100">the</span> <span m="794210">amount</span> <span m="794620">by</span> <span m="794770">which</span> <span m="795010">the</span> <span m="795140">i'th</span> <span m="795390">block</span> <span m="795900">hangs</span> <span m="796330">out</span> <span m="796620">over</span> <span m="796850">the</span> <span m="796970">edge.</span> <span m="810310">and</span> <span m="810420">we're</span> <span m="810480">going</span> <span m="810600">to</span> <span m="810660">count</span> <span m="810990">from</span> <span m="811130">the</span> <span m="811210">top.</span> </p>
<p><span m="825070">All</span> <span m="825210">right.</span> <span m="825330">So</span> <span m="825440">let's</span> <span m="825930">see</span> <span m="826160">this</span> <span m="826370">on</span> <span m="826480">a</span> <span m="826540">picture.</span> <span m="829270">So</span> <span m="829460">I</span> <span m="829530">got</span> <span m="829710">the</span> <span m="829780">table</span> <span m="830620">here.</span> <span m="835220">And</span> <span m="835370">then</span> <span m="835520">I'm</span> <span m="835670">stacking</span> <span m="836150">the</span> <span m="836220">blocks</span> <span m="836670">out</span> <span m="839190">in</span> <span m="839310">some</span> <span m="839600">way</span> <span m="839800">like</span> <span m="840060">this.</span> <span m="842160">And</span> <span m="842350">with</span> <span m="842480">a</span> <span m="842530">greedy</span> <span m="842880">strategy,</span> <span m="843430">you</span> <span m="843500">don't</span> <span m="843750">ever</span> <span m="843910">go</span> <span m="844060">backwards.</span> <span m="852430">All</span> <span m="852590">right.</span> </p>
<p><span m="852900">So I'm</span> <span m="853030">defining</span> <span m="854520">this</span> <span m="854870">distance.</span> <span m="855980">This</span> <span m="856200">is</span> <span m="856350">block</span> <span m="856700">one.</span> <span m="859110">This</span> <span m="859400">distance</span> <span m="860440">here</span> <span m="864160">is</span> <span m="864360">r</span> <span m="864580">1.</span> <span m="866240">This</span> <span m="866550">distance</span> <span m="868870">is</span> <span m="869140">r</span> <span m="869350">2.</span> <span m="871470">This</span> <span m="871730">one</span> <span m="871900">is</span> <span m="872030">r</span> <span m="872200">3--</span> <span m="872570">and</span> <span m="872680">so</span> <span m="872930">forth,</span> <span m="873610">until</span> <span m="873860">you</span> <span m="873930">get</span> <span m="874040">down</span> <span m="874230">to</span> <span m="874300">here.</span> <span m="874710">This</span> <span m="874820">is</span> <span m="874940">r</span> <span m="875170">n,</span> <span m="876810">because</span> <span m="876980">this</span> <span m="877160">is</span> <span m="877270">the</span> <span m="877400">n'th</span> <span m="877700">block,</span> <span m="878080">and</span> <span m="878170">it</span> <span m="878280">extends</span> <span m="878710">out</span> <span m="878900">r</span> <span m="879180">n</span> <span m="879330">from</span> <span m="879430">the</span> <span m="879490">table.</span> </p>
<p><span m="880610">And</span> <span m="880790">this</span> <span m="880970">is</span> <span m="881090">0</span> <span m="881550">here,</span> <span m="882620">which</span> <span m="883100">the</span> <span m="883170">table--</span> <span m="883590">we'll</span> <span m="883700">consider</span> <span m="884060">the</span> <span m="884160">table</span> <span m="884480">to</span> <span m="884560">be</span> <span m="884690">the</span> <span m="884840">n</span> <span m="884980">plus</span> <span m="885220">first</span> <span m="885490">block.</span> <span m="885860">So</span> <span m="886100">r n</span> <span m="886510">plus</span> <span m="886800">1</span> <span m="888020">is</span> <span m="888200">0.</span> <span m="889940">OK.</span> <span m="891330">And</span> <span m="891540">so</span> <span m="891700">using</span> <span m="892100">the</span> <span m="892170">strategy--</span> <span m="892940">the</span> <span m="893030">greedy</span> <span m="893310">strategy--</span> <span m="893820">we</span> <span m="893940">want</span> <span m="894170">to</span> <span m="894230">know</span> <span m="894790">what's</span> <span m="895080">r</span> <span m="895290">1?</span> <span m="896170">We</span> <span m="896280">want</span> <span m="896420">to</span> <span m="896480">compute</span> <span m="896820">that--</span> <span m="897670">if</span> <span m="897780">I</span> <span m="897880">keep</span> <span m="898200">pushing</span> <span m="898990">the</span> <span m="899130">blocks</span> <span m="899530">out</span> <span m="900040">in a</span> <span m="900090">topward</span> <span m="901000">fashion</span> <span m="901430">as</span> <span m="901530">far</span> <span m="901740">as</span> <span m="901860">they'll</span> <span m="901950">go.</span> </p>
<p><span m="902970">Now</span> <span m="903210">the</span> <span m="903310">constraint</span> <span m="904660">is</span> <span m="905210">the</span> <span m="905300">stability</span> <span m="905780">constraint</span> <span m="906850">so</span> <span m="907040">it</span> <span m="907120">won't</span> <span m="907360">fall</span> <span m="907700">over.</span> <span m="915930">The</span> <span m="916030">stability</span> <span m="916490">constraint</span> <span m="917070">says</span> <span m="917690">that</span> <span m="918210">the</span> <span m="918460">center</span> <span m="918840">of</span> <span m="918940">mass--</span> <span m="924970">we'll</span> <span m="925090">call</span> <span m="925340">it</span> <span m="925430">C</span> <span m="925800">k--</span> <span m="927760">of</span> <span m="928000">the</span> <span m="928070">top</span> <span m="928510">k</span> <span m="928730">blocks</span> <span m="937010">must</span> <span m="937390">lie</span> <span m="938730">on</span> <span m="938950">the</span> <span m="939020">k</span> <span m="939220">plus</span> <span m="939480">first</span> <span m="939760">block.</span> <span m="940150">Otherwise</span> <span m="940440">it'll</span> <span m="940600">tip</span> <span m="940830">over.</span> </p>
<p><span m="949090">And</span> <span m="949220">of</span> <span m="949290">course</span> <span m="949540">the</span> <span m="949630">table</span> <span m="950220">is</span> <span m="950320">block</span> <span m="950710">n</span> <span m="950820">plus</span> <span m="951080">1</span> <span m="952630">in</span> <span m="952760">this</span> <span m="953360">context.</span> <span m="962100">OK.</span> <span m="962840">So</span> <span m="963910">if</span> <span m="964070">at</span> <span m="964160">any</span> <span m="964430">point</span> <span m="965990">we've</span> <span m="966200">got</span> <span m="966530">k</span> <span m="966820">blocks,</span> <span m="967550">and</span> <span m="967670">their</span> <span m="967800">center</span> <span m="968130">of</span> <span m="968210">mass</span> <span m="968620">is</span> <span m="970090">out</span> <span m="970340">here,</span> <span m="971200">they'll</span> <span m="971350">fall</span> <span m="971700">over</span> <span m="971950">at</span> <span m="972040">that</span> <span m="972220">point.</span> <span m="973040">Or</span> <span m="973280">if</span> <span m="973370">you</span> <span m="973460">are</span> <span m="973510">building</span> <span m="973810">it</span> <span m="973890">this</span> <span m="974090">way,</span> <span m="974850">they</span> <span m="974960">would</span> <span m="975090">fall</span> <span m="975400">over</span> <span m="975570">that</span> <span m="975790">way.</span> <span m="976690">All</span> <span m="976800">right.</span> <span m="976890">So</span> <span m="977030">for</span> <span m="977140">each</span> <span m="977390">set</span> <span m="977630">of</span> <span m="978070">the</span> <span m="978170">top</span> <span m="978430">k</span> <span m="978590">blocks,</span> <span m="979010">the</span> <span m="979080">center</span> <span m="979310">of</span> <span m="979370">mass</span> <span m="979680">has</span> <span m="980000">to</span> <span m="980080">be</span> <span m="980160">on</span> <span m="980250">the</span> <span m="980320">block</span> <span m="980630">below.</span> <span m="981280">And</span> <span m="981440">if</span> <span m="981520">it</span> <span m="981600">is,</span> <span m="982410">we're</span> <span m="982560">fine.</span> <span m="983300">It</span> <span m="983450">won't</span> <span m="983790">fall</span> <span m="984140">over.</span> </p>
<p><span m="987410">Now</span> <span m="987690">for</span> <span m="987840">the</span> <span m="987930">greedy</span> <span m="988340">stacking.</span> <span m="999540">Where</span> <span m="1000010">is</span> <span m="1000350">the</span> <span m="1000430">center</span> <span m="1000810">of</span> <span m="1000910">mass</span> <span m="1001320">of</span> <span m="1001390">the</span> <span m="1001460">top</span> <span m="1001760">k</span> <span m="1001940">blocks</span> <span m="1002560">in</span> <span m="1002690">the</span> <span m="1002760">greedy</span> <span m="1003140">stacking,</span> <span m="1004380">the</span> <span m="1004490">way</span> <span m="1004630">we</span> <span m="1004730">defined</span> <span m="1005140">them</span> <span m="1007220">in</span> <span m="1007340">terms</span> <span m="1007710">of</span> <span m="1008020">these</span> <span m="1008280">variables.</span> <span m="1009270">What</span> <span m="1009410">is</span> <span m="1009550">C</span> <span m="1009998">k?</span> </p>
<p><span m="1015270">All  right.</span> <span m="1015370">So I</span> <span m="1015580">got</span> <span m="1015720">the</span> <span m="1015790">top</span> <span m="1016150">k</span> <span m="1016370">blocks</span> <span m="1021040">r</span> <span m="1021410">k</span> <span m="1021650">plus</span> <span m="1021960">1.</span> <span m="1022190">The</span> <span m="1022260">center</span> <span m="1022650">of</span> <span m="1022750">mass</span> <span m="1023150">of</span> <span m="1023210">the</span> <span m="1023280">top</span> <span m="1023580">k</span> <span m="1023770">blocks</span> <span m="1024940">by</span> <span m="1025099">the</span> <span m="1025210">greedy</span> <span m="1025569">strategy</span> <span m="1026089">is</span> <span m="1026260">right</span> <span m="1026700">at</span> <span m="1026790">the</span> <span m="1026970">edge</span> <span m="1027510">of</span> <span m="1027619">the</span> <span m="1027740">k'th</span> <span m="1027980">block.</span> <span m="1029349">So</span> <span m="1029510">if</span> <span m="1029579">this</span> <span m="1029760">is</span> <span m="1029829">a</span> <span m="1029859">block</span> <span m="1030194">k</span> <span m="1030530">here,</span> <span m="1031270">then</span> <span m="1031450">this</span> <span m="1032109">is</span> <span m="1032319">r</span> <span m="1032480">k</span> <span m="1032660">plus</span> <span m="1032940">1.</span> </p>
<p><span m="1035240">That's</span> <span m="1035550">where</span> <span m="1035640">the</span> <span m="1035720">center</span> <span m="1036000">of</span> <span m="1036060">mass</span> <span m="1036390">has</span> <span m="1036589">to</span> <span m="1036680">be</span> <span m="1036910">of</span> <span m="1037020">all</span> <span m="1037250">these</span> <span m="1037480">blocks--</span> <span m="1038470">the</span> <span m="1038550">top</span> <span m="1038829">k</span> <span m="1039109">blocks.</span> <span m="1040500">If</span> <span m="1040700">it's</span> <span m="1040819">a</span> <span m="1040869">little</span> <span m="1041109">bit</span> <span m="1041270">more,</span> <span m="1041670">it's</span> <span m="1041770">going</span> <span m="1041890">to</span> <span m="1041950">fall</span> <span m="1042250">over.</span> <span m="1043270">If</span> <span m="1043450">it's</span> <span m="1043609">less,</span> <span m="1044109">then</span> <span m="1044250">it's</span> <span m="1044450">not</span> <span m="1044849">greedy.</span> <span m="1045480">We</span> <span m="1045619">didn't</span> <span m="1045839">push</span> <span m="1046150">it</span> <span m="1046230">out</span> <span m="1046410">as</span> <span m="1046510">far</span> <span m="1046760">as</span> <span m="1046869">it</span> <span m="1046950">could</span> <span m="1047079">go.</span> <span m="1048510">All</span> <span m="1048670">right.</span> </p>
<p><span m="1050010">So</span> <span m="1050180">for</span> <span m="1050270">the</span> <span m="1050350">greedy</span> <span m="1050720">stacking,</span> <span m="1051280">C</span> <span m="1051575">k</span> <span m="1051870">is</span> <span m="1052010">equal to</span> <span m="1052390">r</span> <span m="1052760">k plus</span> <span m="1053020">1.</span> <span m="1056550">And</span> <span m="1058970">r</span> <span m="1059315">n</span> <span m="1059660">plus</span> <span m="1060000">1</span> <span m="1060340">of</span> <span m="1060420">course</span> <span m="1060720">is</span> <span m="1060820">the</span> <span m="1060930">edge</span> <span m="1061080">of</span> <span m="1061150">the</span> <span m="1061210">table.</span> <span m="1062090">That's</span> <span m="1062350">zero.</span> </p>
<p><span m="1065880">Any</span> <span m="1066070">questions</span> <span m="1066510">so</span> <span m="1066660">far?</span> <span m="1068540">What</span> <span m="1068690">we're</span> <span m="1068790">doing?</span> <span m="1070500">OK.</span> </p>
<p><span m="1071570">All</span> <span m="1071860">right.</span> <span m="1072080">So</span> <span m="1072410">let's</span> <span m="1072670">figure</span> <span m="1072960">out</span> <span m="1073180">where</span> <span m="1073290">the</span> <span m="1073390">center</span> <span m="1073790">of</span> <span m="1073890">mass</span> <span m="1074410">of</span> <span m="1074510">the</span> <span m="1074580">top</span> <span m="1074890">k</span> <span m="1075180">blocks</span> <span m="1075440">is--</span> <span m="1076190">a</span> <span m="1076460">formula</span> <span m="1076930">for</span> <span m="1077050">that--</span> <span m="1077360">a</span> <span m="1077480">recursive</span> <span m="1077900">formula.</span> <span m="1079600">So</span> <span m="1080670">to</span> <span m="1080800">do</span> <span m="1081000">that</span> <span m="1082000">we</span> <span m="1082160">look</span> <span m="1082330">at</span> <span m="1082400">the</span> <span m="1082470">center</span> <span m="1082790">of</span> <span m="1082880">mass</span> <span m="1083280">of</span> <span m="1083400">the</span> <span m="1083470">k'th</span> <span m="1083920">block</span> <span m="1084260">by</span> <span m="1084430">itself,</span> <span m="1085800">and</span> <span m="1085920">then</span> <span m="1086050">average</span> <span m="1086400">it</span> <span m="1086480">in</span> <span m="1086820">with</span> <span m="1086970">the</span> <span m="1087050">center</span> <span m="1087360">of</span> <span m="1087440">mass</span> <span m="1087900">of</span> <span m="1087990">the</span> <span m="1088060">top</span> <span m="1088470">k</span> <span m="1088720">minus</span> <span m="1089070">1</span> <span m="1089290">blocks.</span> <span m="1090710">OK.</span> <span m="1091050">So</span> <span m="1091140">we're</span> <span m="1091200">going</span> <span m="1091320">to</span> <span m="1091390">set</span> <span m="1091560">up</span> <span m="1091660">a</span> <span m="1091720">recursion</span> <span m="1092320">to</span> <span m="1092770">compute</span> <span m="1093140">C</span> <span m="1093624">k.</span> </p>
<p><span m="1095560">Where</span> <span m="1095920">is</span> <span m="1096070">the</span> <span m="1096170">center</span> <span m="1096580">of</span> <span m="1096690">mass</span> <span m="1098130">of</span> <span m="1098600">the</span> <span m="1098890">k'th</span> <span m="1099420">block</span> <span m="1102220">just</span> <span m="1102510">by</span> <span m="1102700">itself?</span> <span m="1103150">The</span> <span m="1103250">center</span> <span m="1103580">of</span> <span m="1103650">mass</span> <span m="1103990">of</span> <span m="1104070">the</span> <span m="1104140">k'th</span> <span m="1104490">block.</span> <span m="1106480">This</span> <span m="1106600">guy.</span> <span m="1106780">Where is</span> <span m="1107110">its</span> <span m="1107270">center</span> <span m="1107520">of</span> <span m="1107610">mass?</span> </p>
<p><span m="1109970">AUDIENCE: [INAUDIBLE]</span> </p>
<p></p>
<p><span m="1112670">PROFESSOR: Yeah,</span> <span m="1113010">r</span> <span m="1113250">k</span> <span m="1113490">plus</span> <span m="1113870">1</span> <span m="1114540">minus</span> <span m="1114920">0.5.</span> <span m="1115840">Because</span> <span m="1116820">the</span> <span m="1116940">center</span> <span m="1117260">of</span> <span m="1117360">mass</span> <span m="1117550">of</span> <span m="1117610">this</span> <span m="1117770">block</span> <span m="1118200">is</span> <span m="1118330">1/2</span> <span m="1119820">from</span> <span m="1120150">the</span> <span m="1120250">right</span> <span m="1120480">edge.</span> <span m="1120860">And</span> <span m="1120950">the</span> <span m="1121030">right</span> <span m="1121300">edge</span> <span m="1122280">is--</span> <span m="1123380">oh,</span> <span m="1123540">it's</span> <span m="1123690">r</span> <span m="1123880">k.</span> <span m="1125450">Sorry.</span> <span m="1126315">All</span> <span m="1126690">right?</span> <span m="1126900">The</span> <span m="1126970">top.</span> <span m="1127590">Yeah</span> <span m="1127800">this</span> <span m="1128000">is.</span> <span m="1128850">What?</span> <span m="1129060">What</span> <span m="1129180">did I</span> <span m="1129320">do</span> <span m="1129530">here?</span> <span m="1129800">Oh,</span> <span m="1129910">the</span> <span m="1129990">top</span> <span m="1130310">k.</span> <span m="1130610">Sorry.</span> <span m="1130950">This</span> <span m="1131120">is</span> <span m="1131190">the</span> <span m="1131270">k'th</span> <span m="1131510">block,</span> <span m="1131850">the</span> <span m="1131910">way</span> <span m="1132030">I</span> <span m="1132130">had</span> <span m="1132290">it.</span> <span m="1133080">Here's</span> <span m="1133300">the</span> <span m="1133380">k'th.</span> <span m="1133740">So</span> <span m="1133930">it</span> <span m="1134020">is</span> <span m="1134270">r</span> <span m="1134680">k minus</span> <span m="1134990">a</span> <span m="1135080">half.</span> <span m="1136520">That's</span> <span m="1136800">right.</span> <span m="1137060">This</span> <span m="1137210">is</span> <span m="1137320">r</span> <span m="1137440">k</span> <span m="1137630">plus</span> <span m="1137860">1</span> <span m="1138350">down here.</span> </p>
<p><span m="1139330">All</span> <span m="1139470">right.</span> <span m="1139660">Because</span> <span m="1140230">the</span> <span m="1140360">right</span> <span m="1140650">edge</span> <span m="1140860">of</span> <span m="1140930">the</span> <span m="1141000">k'th</span> <span m="1141260">block</span> <span m="1141560">is</span> <span m="1141650">at</span> <span m="1141760">r</span> <span m="1142245">k,</span> <span m="1142730">the</span> <span m="1142830">center</span> <span m="1143110">of</span> <span m="1143210">mass</span> <span m="1143550">is</span> <span m="1143670">1/2</span> <span m="1144070">left</span> <span m="1144750">of</span> <span m="1144900">that.</span> <span m="1146400">All</span> <span m="1146620">right.</span> <span m="1146820">So</span> <span m="1146950">we</span> <span m="1147060">know</span> <span m="1147370">that</span> <span m="1147550">the</span> <span m="1148240">center</span> <span m="1148620">of</span> <span m="1148720">mass</span> <span m="1154750">of</span> <span m="1154870">the</span> <span m="1154950">k'th</span> <span m="1155330">block</span> <span m="1159040">is</span> <span m="1159430">at</span> <span m="1160570">r</span> <span m="1160860">k</span> <span m="1161150">minus</span> <span m="1162000">1/2.</span> </p>
<p><span m="1164180">All</span> <span m="1164640">right.</span> <span m="1164870">Now</span> <span m="1165130">we</span> <span m="1165230">can</span> <span m="1165370">compute</span> <span m="1166360">another</span> <span m="1166710">expression</span> <span m="1167280">for</span> <span m="1167410">C</span> <span m="1167695">k.</span> <span m="1170520">The</span> <span m="1172790">center</span> <span m="1174000">of</span> <span m="1174210">mass</span> <span m="1178840">of</span> <span m="1180050">the</span> <span m="1180670">top</span> <span m="1182032">k</span> <span m="1182420">blocks</span> <span m="1185610">is</span> <span m="1188120">C</span> <span m="1188550">k</span> <span m="1189610">equals--</span> <span m="1191230">well</span> <span m="1191360">there's</span> <span m="1191750">k</span> <span m="1191930">minus</span> <span m="1192310">1</span> <span m="1192620">block</span> <span m="1193050">centered</span> <span m="1193660">at</span> <span m="1193850">C</span> <span m="1194140">k</span> <span m="1194430">minus</span> <span m="1194800">1,</span> <span m="1196120">because</span> <span m="1196290">the</span> <span m="1196370">center</span> <span m="1196640">of</span> <span m="1196710">mass</span> <span m="1196990">of</span> <span m="1197050">the</span> <span m="1197110">top</span> <span m="1197390">k</span> <span m="1197550">minus</span> <span m="1197860">1</span> <span m="1198030">block's</span> <span m="1198280">by</span> <span m="1198400">definition</span> <span m="1198675">is</span> <span m="1198950">C</span> <span m="1198990">k</span> <span m="1199350">minus</span> <span m="1199670">1.</span> <span m="1200180">And</span> <span m="1200360">they</span> <span m="1200470">have</span> <span m="1200660">weight</span> <span m="1201140">k</span> <span m="1201320">minus</span> <span m="1201650">1.</span> <span m="1202910">And</span> <span m="1203070">then</span> <span m="1203160">we</span> <span m="1203290">average</span> <span m="1203700">that</span> <span m="1203840">with</span> <span m="1203910">a</span> <span m="1203990">k'th</span> <span m="1204330">block</span> <span m="1204610">that</span> <span m="1204700">has</span> <span m="1204880">weight</span> <span m="1205160">1</span> <span m="1205700">at</span> <span m="1205860">position</span> <span m="1206470">r</span> <span m="1206685">k</span> <span m="1206900">minus</span> <span m="1207240">1/2.</span> <span m="1208930">And</span> <span m="1209120">the</span> <span m="1209190">total</span> <span m="1209720">weight</span> <span m="1210180">of</span> <span m="1210310">this</span> <span m="1210810">is</span> <span m="1211760">k</span> <span m="1212070">minus</span> <span m="1212520">1</span> <span m="1212760">here,</span> <span m="1213980">and</span> <span m="1214200">one</span> <span m="1214430">for</span> <span m="1214560">this</span> <span m="1214760">block.</span> </p>
<p><span m="1216160">So</span> <span m="1216290">this</span> <span m="1216540">equals</span> <span m="1217580">KC</span> <span m="1218260">k</span> <span m="1218600">minus</span> <span m="1219000">1</span> <span m="1221450">minus--</span> <span m="1224160">let me</span> <span m="1224330">write</span> <span m="1224480">it</span> <span m="1224780">better</span> <span m="1225020">here.</span> <span m="1225600">k</span> <span m="1225810">minus</span> <span m="1226300">1.</span> <span m="1227860">C</span> <span m="1228250">k</span> <span m="1228640">minus</span> <span m="1229060">1</span> <span m="1230500">plus</span> <span m="1231030">R</span> <span m="1231430">k</span> <span m="1232170">minus</span> <span m="1232630">1/2</span> <span m="1233680">all</span> <span m="1234050">over</span> <span m="1234320">k.</span> <span m="1236350">All right.</span> <span m="1236640">So now</span> <span m="1236850">we've</span> <span m="1236970">got</span> <span m="1237100">an</span> <span m="1237180">expression--</span> <span m="1238560">a</span> <span m="1238640">recursive</span> <span m="1239110">expression--</span> <span m="1239670">for</span> <span m="1239810">C</span> <span m="1240295">k.</span> </p>
<p><span m="1244820">And</span> <span m="1245830">to</span> <span m="1245950">simplify</span> <span m="1246640">it,</span> <span m="1246840">we're</span> <span m="1246940">going</span> <span m="1246960">to</span> <span m="1247040">plug in</span> <span m="1247530">that--</span> <span m="1248000">C k</span> <span m="1248425">as r</span> <span m="1248850">k</span> <span m="1249020">plus</span> <span m="1249300">1</span> <span m="1249570">for</span> <span m="1249670">the</span> <span m="1249750">greedy</span> <span m="1250000">strategy.</span> <span m="1250470">Yeah?</span> </p>
<p><span m="1252104">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="1254570">PROFESSOR: L</span> <span m="1254780">is</span> <span m="1254900">1.</span> <span m="1255900">The</span> <span m="1256040">length</span> <span m="1256410">of</span> <span m="1256480">the</span> <span m="1256540">blocks</span> <span m="1256970">is</span> <span m="1257090">1,</span> <span m="1257360">here</span> <span m="1258630">to</span> <span m="1258750">make</span> <span m="1258990">it</span> <span m="1259090">simpler.</span> <span m="1259530">Otherwise</span> <span m="1260000">it</span> <span m="1260090">would</span> <span m="1260220">have</span> <span m="1260310">been</span> <span m="1260400">a</span> <span m="1260460">half</span> <span m="1260750">L.</span> <span m="1261010">So</span> <span m="1261140">yesterday</span> <span m="1261670">we</span> <span m="1261790">talked</span> <span m="1262110">about</span> <span m="1262450">length</span> <span m="1262795">L</span> <span m="1263140">blocks.</span> <span m="1263890">Today</span> <span m="1264250">they're</span> <span m="1264370">length</span> <span m="1264680">1,</span> <span m="1265160">to</span> <span m="1265270">be</span> <span m="1265370">simple.</span> </p>
<p><span m="1267020">All</span> <span m="1267220">right.</span> <span m="1267370">So</span> <span m="1267470">now</span> <span m="1267660">for</span> <span m="1267770">the</span> <span m="1267840">greedy</span> <span m="1268170">strategy,</span> <span m="1268660">I'm</span> <span m="1268750">just</span> <span m="1268900">going</span> <span m="1269020">to</span> <span m="1269090">plug</span> <span m="1269365">in</span> <span m="1269640">C k</span> <span m="1270130">as r</span> <span m="1270620">k</span> <span m="1270830">plus</span> <span m="1271150">1.</span> <span m="1272580">So</span> <span m="1272750">let's</span> <span m="1272970">do</span> <span m="1273120">that.</span> <span m="1275680">So</span> <span m="1275850">I</span> <span m="1275910">got</span> <span m="1276120">C</span> <span m="1276455">k</span> <span m="1276790">as</span> <span m="1277355">r</span> <span m="1277620">k</span> <span m="1278090">plus</span> <span m="1278430">1</span> <span m="1278980">equals</span> <span m="1281780">zero</span> <span m="1282400">k</span> <span m="1282960">minus</span> <span m="1283440">1.</span> <span m="1284830">Well</span> <span m="1285250">C k</span> <span m="1285750">minus</span> <span m="1286140">1</span> <span m="1286340">is</span> <span m="1286450">just</span> <span m="1286750">R</span> <span m="1287035">k</span> <span m="1291540">plus</span> <span m="1292120">R</span> <span m="1292520">k</span> <span m="1292920">minus</span> <span m="1293280">1/2</span> <span m="1295570">over</span> <span m="1295900">K.</span> <span m="1297090">And</span> <span m="1297350">this</span> <span m="1297540">simplifies</span> <span m="1299130">to</span> <span m="1300300">k r</span> <span m="1300720">k</span> <span m="1302270">minus</span> <span m="1302660">r</span> <span m="1302990">k plus</span> <span m="1303270">r k--</span> <span m="1303700">they cancel--</span> <span m="1304390">minus</span> <span m="1304760">1/2</span> <span m="1305850">over</span> <span m="1306110">k.</span> <span m="1309010">This</span> <span m="1309350">equals</span> <span m="1310010">r</span> <span m="1310395">k</span> <span m="1311070">minus</span> <span m="1311480">1/2.</span> <span m="1312790">So</span> <span m="1312970">getting</span> <span m="1313210">very</span> <span m="1313550">simple</span> <span m="1314410">now.</span> <span m="1316020">And</span> <span m="1316380">I'm</span> <span m="1316490">just</span> <span m="1316620">going</span> <span m="1316740">to</span> <span m="1316820">rewrite</span> <span m="1317370">this</span> <span m="1317780">to</span> <span m="1318250">look</span> <span m="1318460">at</span> <span m="1318550">the</span> <span m="1318620">difference</span> <span m="1319230">between</span> <span m="1320560">r k</span> <span m="1320995">and r</span> <span m="1321430">k</span> <span m="1321660">plus</span> <span m="1321960">1.</span> <span m="1322950">So</span> <span m="1323150">r</span> <span m="1323405">k</span> <span m="1323660">minus</span> <span m="1324210">r k</span> <span m="1324650">plus</span> <span m="1325010">1</span> <span m="1327175">Oops.</span> <span m="1327610">I</span> <span m="1327670">made</span> <span m="1327790">a</span> <span m="1327840">mistake.</span> <span m="1328270">That's</span> <span m="1328450">1</span> <span m="1328610">over</span> <span m="1328760">2</span> <span m="1329000">k.</span> <span m="1329700">Half</span> <span m="1329970">over</span> <span m="1330180">k</span> <span m="1331300">is</span> <span m="1331590">1</span> <span m="1331760">over</span> <span m="1331910">2K.</span> </p>
<p><span m="1336500">Any</span> <span m="1336680">questions</span> <span m="1338230">about</span> <span m="1338540">the</span> <span m="1338600">math</span> <span m="1338880">we</span> <span m="1338970">did?</span> <span m="1341080">Interesting.</span> <span m="1341420">This</span> <span m="1341560">is</span> <span m="1341660">one</span> <span m="1341790">of</span> <span m="1341850">those</span> <span m="1342010">problems</span> <span m="1342530">where</span> <span m="1342650">the</span> <span m="1342720">math</span> <span m="1343390">this</span> <span m="1343700">way</span> <span m="1343970">is</span> <span m="1344110">easy.</span> <span m="1345390">There</span> <span m="1345570">are</span> <span m="1345720">other</span> <span m="1345890">ways</span> <span m="1346140">to</span> <span m="1346210">try</span> <span m="1346350">to</span> <span m="1346430">solve</span> <span m="1346690">this</span> <span m="1346830">problem</span> <span m="1347140">and the</span> <span m="1347200">math</span> <span m="1347480">gets</span> <span m="1347670">pretty</span> <span m="1347890">hairy.</span> <span m="1349380">But</span> <span m="1349500">this</span> <span m="1349660">one's</span> <span m="1349880">a</span> <span m="1350180">simple</span> <span m="1350570">approach.</span> <span m="1352220">What's</span> <span m="1352670">the</span> <span m="1352760">meaning</span> <span m="1353090">of</span> <span m="1353170">this?</span> <span m="1354030">What's</span> <span m="1354270">an</span> <span m="1354330">interpretation</span> <span m="1355140">of</span> <span m="1355260">r k</span> <span m="1355660">minus</span> <span m="1356000">r k</span> <span m="1356280">plus</span> <span m="1356540">1?</span> <span m="1359090">Yeah.</span> </p>
<p><span m="1360082">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="1363290">PROFESSOR: Yeah.</span> <span m="1363590">This</span> <span m="1363890">is</span> <span m="1364070">how</span> <span m="1364290">much</span> <span m="1364550">further</span> <span m="1364950">the</span> <span m="1365060">k'th</span> <span m="1365490">block</span> <span m="1365910">sticks</span> <span m="1366270">out</span> <span m="1366930">over</span> <span m="1367270">the</span> <span m="1367360">k</span> <span m="1367620">plus</span> <span m="1367910">first</span> <span m="1368210">block.</span> <span m="1369160">So</span> <span m="1369320">let's</span> <span m="1369510">see</span> <span m="1369750">that.</span> <span m="1371398">All</span> <span m="1371810">right.</span> <span m="1372170">So</span> <span m="1373050">here's</span> <span m="1373400">r</span> <span m="1373660">1.</span> <span m="1374230">Here's</span> <span m="1374500">r</span> <span m="1374750">2.</span> <span m="1376020">r</span> <span m="1376295">1</span> <span m="1376570">minus</span> <span m="1377070">r</span> <span m="1377445">2</span> <span m="1378150">is</span> <span m="1378470">the</span> <span m="1378560">amount</span> <span m="1378900">by</span> <span m="1379030">which</span> <span m="1379380">the</span> <span m="1379450">first</span> <span m="1379780">block</span> <span m="1380070">sticks</span> <span m="1380350">out</span> <span m="1380490">over</span> <span m="1380620">the</span> <span m="1380700">second.</span> <span m="1382010">Here's</span> <span m="1382440">r 2</span> <span m="1382680">two</span> <span m="1383050">minus</span> <span m="1383480">r 3.</span> <span m="1384700">It's</span> <span m="1384900">that</span> <span m="1385130">distance.</span> <span m="1387000">All</span> <span m="1387130">right.</span> <span m="1387340">So</span> <span m="1387450">we've</span> <span m="1387630">gotten</span> <span m="1387970">a</span> <span m="1388020">formula</span> <span m="1388600">for</span> <span m="1388780">the</span> <span m="1388860">greedy</span> <span m="1389200">strategy</span> <span m="1390580">of</span> <span m="1391260">how</span> <span m="1391510">much</span> <span m="1391760">the</span> <span m="1391840">k'th</span> <span m="1392230">block</span> <span m="1392560">sticks</span> <span m="1392970">out</span> <span m="1393730">over</span> <span m="1393970">the</span> <span m="1394060">k</span> <span m="1394280">plus</span> <span m="1394550">first</span> <span m="1394830">block.</span> <span m="1395630">And</span> <span m="1395750">it's</span> <span m="1395850">pretty</span> <span m="1396060">simple.</span> </p>
<p><span m="1397420">All</span> <span m="1397610">right.</span> <span m="1397810">So</span> <span m="1397940">now</span> <span m="1399340">we</span> <span m="1399480">can</span> <span m="1399640">write</span> <span m="1399890">down</span> <span m="1401930">this</span> <span m="1402180">expression</span> <span m="1402740">for</span> <span m="1402920">all</span> <span m="1403170">the</span> <span m="1403830">values</span> <span m="1404160">of</span> <span m="1404240">k.</span> <span m="1405340">So</span> <span m="1405530">what's</span> <span m="1405760">r 1</span> <span m="1406240">minus r 2</span> <span m="1407582">going</span> <span m="1408080">to</span> <span m="1408150">be</span> <span m="1409680">in the</span> <span m="1409790">greedy</span> <span m="1410140">strategy?</span> <span m="1410620">How</span> <span m="1410830">much</span> <span m="1411110">is</span> <span m="1411230">the</span> <span m="1411310">top</span> <span m="1411660">block</span> <span m="1411940">sticking</span> <span m="1412320">out</span> <span m="1412480">over</span> <span m="1412630">the</span> <span m="1412720">second</span> <span m="1413040">block</span> <span m="1415970">in</span> <span m="1416060">the</span> <span m="1416140">greedy</span> <span m="1416550">strategy?</span> <span m="1418130">1/2.</span> <span m="1418860">Just</span> <span m="1419060">plug in</span> <span m="1419390">k</span> <span m="1419710">equals</span> <span m="1419880">1.</span> <span m="1422140">r</span> <span m="1422380">2</span> <span m="1422710">minus</span> <span m="1423070">r</span> <span m="1423230">3</span> <span m="1423840">equals--</span> </p>
<p><span m="1426758">AUDIENCE: A quarter.</span> <span m="1429105">One</span> <span m="1429460">quarter.</span> <span m="1430880">r</span> <span m="1431190">3</span> <span m="1431670">minus</span> <span m="1432090">r</span> <span m="1432430">4</span> <span m="1434890">is</span> <span m="1435130">what?</span> <span m="1436430">6.</span> <span m="1437270">All</span> <span m="1437690">right.</span> <span m="1437950">So</span> <span m="1438060">now</span> <span m="1438200">you</span> <span m="1438330">can</span> <span m="1438460">start</span> <span m="1438760">to</span> <span m="1438830">see</span> <span m="1439020">what</span> <span m="1439180">the</span> <span m="1439250">TAs</span> <span m="1439690">did,</span> <span m="1440030">how</span> <span m="1440190">it</span> <span m="1440290">got</span> <span m="1440480">smaller</span> <span m="1441250">each</span> <span m="1441460">step</span> <span m="1441750">going</span> <span m="1442020">down</span> <span m="1442990">all</span> <span m="1443240">the</span> <span m="1443320">way</span> <span m="1443550">to</span> <span m="1443670">the</span> <span m="1443820">end</span> <span m="1444060">here.</span> <span m="1444250">You</span> <span m="1444350">get</span> <span m="1444590">r</span> <span m="1445680">n--</span> <span m="1446440">the</span> <span m="1446570">n'th</span> <span m="1446740">block--</span> <span m="1447060">minus</span> <span m="1447410">the</span> <span m="1448220">table</span> <span m="1449650">is</span> <span m="1450130">1</span> <span m="1450400">over</span> <span m="1450630">2</span> <span m="1450820">n.</span> </p>
<p><span m="1453326">All</span> <span m="1453700">right.</span> <span m="1454010">Now</span> <span m="1454170">there's</span> <span m="1454380">a</span> <span m="1454420">very</span> <span m="1454750">easy</span> <span m="1455100">way</span> <span m="1455280">to</span> <span m="1455350">figure</span> <span m="1455710">out</span> <span m="1455980">what</span> <span m="1456490">r</span> <span m="1456960">1</span> <span m="1457430">is,</span> <span m="1458600">which</span> <span m="1458860">is</span> <span m="1459370">what</span> <span m="1459530">we're</span> <span m="1459660">after.</span> <span m="1459970">How</span> <span m="1460230">far</span> <span m="1460470">out</span> <span m="1460600">is</span> <span m="1460690">the</span> <span m="1460760">top</span> <span m="1461010">block?</span> <span m="1461630">What</span> <span m="1461900">do</span> <span m="1462010">I</span> <span m="1462100">do?</span> <span m="1464880">Add them</span> <span m="1465250">all</span> <span m="1465460">together.</span> <span m="1468170">And</span> <span m="1468600">everything</span> <span m="1469010">cancels.</span> <span m="1471520">So</span> <span m="1471600">I</span> <span m="1471670">get</span> <span m="1471920">r 1</span> <span m="1472620">minus</span> <span m="1473100">r</span> <span m="1473580">n plus</span> <span m="1474060">1</span> <span m="1475220">equals</span> <span m="1477180">1/2</span> <span m="1478000">plus</span> <span m="1478310">1/4</span> <span m="1479430">plus</span> <span m="1479830">1/6</span> <span m="1481650">out to</span> <span m="1482105">1</span> <span m="1483015">over</span> <span m="1483470">2 n.</span> <span m="1487010">All</span> <span m="1487130">right.</span> <span m="1487350">So</span> <span m="1487460">that's</span> <span m="1487710">just</span> <span m="1487980">equal</span> <span m="1488260">to</span> <span m="1488430">the</span> <span m="1488520">sum</span> <span m="1489880">i</span> <span m="1490180">equal</span> <span m="1490650">1</span> <span m="1491565">to</span> <span m="1491890">n</span> <span m="1492420">1</span> <span m="1492610">over</span> <span m="1492810">2</span> <span m="1493230">1.</span> </p>
<p><span m="1495330">All</span> <span m="1495490">right.</span> <span m="1495710">Now</span> <span m="1497600">I</span> <span m="1497700">got</span> <span m="1497880">r 1</span> <span m="1498270">minus</span> <span m="1498600">r</span> <span m="1498730">n</span> <span m="1498820">plus</span> <span m="1499070">1.</span> <span m="1499990">But</span> <span m="1500390">what's</span> <span m="1500660">r</span> <span m="1500800">n</span> <span m="1500890">plus</span> <span m="1501120">1?</span> <span m="1503810">0.</span> <span m="1504350">That's</span> <span m="1504560">the</span> <span m="1504640">table.</span> <span m="1505080">The</span> <span m="1505170">table</span> <span m="1505470">doesn't</span> <span m="1505710">hang</span> <span m="1505950">out</span> <span m="1506070">over</span> <span m="1506240">itself.</span> <span m="1506980">So</span> <span m="1507170">in</span> <span m="1507250">fact</span> <span m="1507630">r</span> <span m="1507910">1</span> <span m="1511210">equals</span> <span m="1511800">a</span> <span m="1511860">half</span> <span m="1513190">of</span> <span m="1513390">the</span> <span m="1513460">sum</span> <span m="1515320">of</span> <span m="1515630">the</span> <span m="1515820">inverses</span> <span m="1516320">of</span> <span m="1516400">the</span> <span m="1516630">first</span> <span m="1517080">n</span> <span m="1517270">integers.</span> </p>
<p><span m="1520130">OK?</span> <span m="1521460">And</span> <span m="1521600">so</span> <span m="1521700">now</span> <span m="1521870">all</span> <span m="1522150">we</span> <span m="1522260">have</span> <span m="1522350">to</span> <span m="1522430">do</span> <span m="1523600">is</span> <span m="1523930">to</span> <span m="1524010">compute</span> <span m="1524520">this</span> <span m="1524710">sum</span> <span m="1525760">to</span> <span m="1525890">know</span> <span m="1526050">how</span> <span m="1526180">far</span> <span m="1526430">out</span> <span m="1526600">it</span> <span m="1526700">is.</span> <span m="1528350">In</span> <span m="1528460">fact,</span> <span m="1528690">this</span> <span m="1528960">sum</span> <span m="1529630">comes</span> <span m="1529910">up</span> <span m="1530020">so</span> <span m="1530400">often</span> <span m="1531090">in</span> <span m="1531260">mathematics</span> <span m="1531860">and</span> <span m="1531950">computer</span> <span m="1532300">science,</span> <span m="1532750">it</span> <span m="1532860">has</span> <span m="1533020">a</span> <span m="1533080">special</span> <span m="1533450">name.</span> <span m="1534260">It's</span> <span m="1534400">called</span> <span m="1534670">the</span> <span m="1534750">harmonic</span> <span m="1535320">sum.</span> <span m="1536150">And</span> <span m="1536350">the</span> <span m="1536420">numbers</span> <span m="1536980">are</span> <span m="1537050">called</span> <span m="1537380">the</span> <span m="1537460">harmonic</span> <span m="1538220">numbers.</span> <span m="1539790">And</span> <span m="1539920">in</span> <span m="1539990">particular,</span> <span m="1540570">the</span> <span m="1540720">n'th</span> <span m="1541040">harmonic</span> <span m="1541510">number</span> <span m="1549860">is</span> <span m="1550280">typically</span> <span m="1550790">denoted</span> <span m="1551390">H</span> <span m="1551880">n</span> <span m="1553200">just</span> <span m="1553560">equals</span> <span m="1553990">the</span> <span m="1554070">sum</span> <span m="1554470">of</span> <span m="1554540">the</span> <span m="1554600">n</span> <span m="1555400">reciprocals.</span> </p>
<p><span m="1560450">All</span> <span m="1560610">right.</span> <span m="1560810">So</span> <span m="1560960">for</span> <span m="1561080">example,</span> <span m="1562380">the</span> <span m="1562480">first</span> <span m="1562810">harmonic</span> <span m="1563260">number</span> <span m="1563570">is</span> <span m="1563680">1.</span> <span m="1564810">The</span> <span m="1564920">second</span> <span m="1565530">is</span> <span m="1565890">1</span> <span m="1566210">plus</span> <span m="1566490">1/2</span> <span m="1567930">equals</span> <span m="1568430">3/2.</span> <span m="1570510">The</span> <span m="1570610">third</span> <span m="1571420">is</span> <span m="1572790">3/2</span> <span m="1573860">plus</span> <span m="1574090">1/3,</span> <span m="1576230">which</span> <span m="1576610">is</span> <span m="1579330">11/6.</span> <span m="1581130">The</span> <span m="1581470">fourth</span> <span m="1581810">harmonic</span> <span m="1582310">number</span> <span m="1583690">is</span> <span m="1583960">that</span> <span m="1584920">plus</span> <span m="1585190">1/4.</span> </p>
<p><span m="1587340">22</span> <span m="1587950">plus</span> <span m="1588200">3/12</span> <span m="1589008">is</span> <span m="1589412">25/12.</span> <span m="1592460">That's</span> <span m="1592690">sort</span> <span m="1592920">of</span> <span m="1592990">an</span> <span m="1593070">interesting</span> <span m="1593650">number,</span> <span m="1593990">because</span> <span m="1594380">it's</span> <span m="1594580">bigger</span> <span m="1594850">than</span> <span m="1595010">2.</span> <span m="1597380">What</span> <span m="1597520">does</span> <span m="1597670">that</span> <span m="1597880">tell</span> <span m="1598120">us?</span> <span m="1600070">Why</span> <span m="1600320">is</span> <span m="1600420">that</span> <span m="1600590">interesting</span> <span m="1600950">from</span> <span m="1601130">this</span> <span m="1601300">perspective</span> <span m="1601565">of</span> <span m="1601830">what</span> <span m="1602030">we're</span> <span m="1602130">doing</span> <span m="1602420">here?</span> </p>
<p><span m="1604790">Now</span> <span m="1605170">you</span> <span m="1605320">can</span> <span m="1605470">get</span> <span m="1605620">a</span> <span m="1605660">block</span> <span m="1606020">off</span> <span m="1606210">the</span> <span m="1606300">table.</span> <span m="1607070">Because</span> <span m="1607550">R1,</span> <span m="1608270">the</span> <span m="1608330">distance</span> <span m="1608860">that</span> <span m="1609010">the</span> <span m="1609210">top</span> <span m="1609520">block</span> <span m="1609830">hangs</span> <span m="1610100">out</span> <span m="1610230">off</span> <span m="1610310">the</span> <span m="1610390">table,</span> <span m="1610800">is</span> <span m="1610910">half</span> <span m="1612160">the</span> <span m="1612280">harmonic</span> <span m="1612730">number.</span> <span m="1613690">Half</span> <span m="1614200">of</span> <span m="1614330">H</span> <span m="1614580">4</span> <span m="1615590">is</span> <span m="1615730">a</span> <span m="1615790">little</span> <span m="1615980">bit</span> <span m="1616110">bigger</span> <span m="1616440">than</span> <span m="1616600">1.</span> <span m="1617930">Which</span> <span m="1618140">means</span> <span m="1618540">with</span> <span m="1618690">four</span> <span m="1618990">blocks</span> <span m="1619950">the</span> <span m="1620060">top</span> <span m="1620420">block</span> <span m="1620670">can</span> <span m="1621080">be</span> <span m="1621210">off</span> <span m="1621390">the</span> <span m="1621480">table.</span> </p>
<p><span m="1622580">Now</span> <span m="1622700">the</span> <span m="1622790">TAs</span> <span m="1623230">did</span> <span m="1623390">it</span> <span m="1623500">using</span> <span m="1623790">five.</span> <span m="1624880">I</span> <span m="1625040">didn't</span> <span m="1625180">even</span> <span m="1625350">get</span> <span m="1625470">it</span> <span m="1625540">to</span> <span m="1625650">work</span> <span m="1625820">with</span> <span m="1625930">five.</span> <span m="1626780">In</span> <span m="1626910">theory</span> <span m="1628020">it'll</span> <span m="1628140">work</span> <span m="1628470">with</span> <span m="1628590">12.</span> <span m="1628950">You</span> <span m="1629030">can</span> <span m="1629160">hang</span> <span m="1629460">one</span> <span m="1629650">out</span> <span m="1629860">there</span> <span m="1630070">just</span> <span m="1630360">a</span> <span m="1630430">24th</span> <span m="1631380">beyond</span> <span m="1631770">the</span> <span m="1631860">edge</span> <span m="1632020">of</span> <span m="1632080">the</span> <span m="1632150">table</span> <span m="1633100">in</span> <span m="1633240">theory.</span> </p>
<p><span m="1635230">All</span> <span m="1635400">right.</span> <span m="1635540">If</span> <span m="1635660">we</span> <span m="1636680">kept</span> <span m="1636960">on</span> <span m="1637170">going,</span> <span m="1638860">h</span> <span m="1639180">of</span> <span m="1639260">a</span> <span m="1639330">million--</span> <span m="1639920">the</span> <span m="1639970">millionth</span> <span m="1640430">harmonic</span> <span m="1640960">number,</span> <span m="1641370">if</span> <span m="1641480">you</span> <span m="1641640">were</span> <span m="1641770">to</span> <span m="1641840">go</span> <span m="1642000">compute</span> <span m="1642420">it,</span> <span m="1643490">is</span> <span m="1643820">14.3927</span> <span m="1646930">and</span> <span m="1647080">some</span> <span m="1647220">other</span> <span m="1647410">stuff.</span> <span m="1649000">So</span> <span m="1649160">if</span> <span m="1649240">I</span> <span m="1649330">had</span> <span m="1649500">a</span> <span m="1649540">million</span> <span m="1650140">blocks</span> <span m="1650670">stacked</span> <span m="1651200">up,</span> <span m="1651620">in</span> <span m="1651810">theory</span> <span m="1652910">how</span> <span m="1653220">far</span> <span m="1653570">would</span> <span m="1653700">I</span> <span m="1653750">get</span> <span m="1653900">the</span> <span m="1653980">top</span> <span m="1654280">block</span> <span m="1654660">off</span> <span m="1654780">the</span> <span m="1654860">table?</span> <span m="1657230">Seven</span> <span m="1657770">blocks.</span> <span m="1659160">All</span> <span m="1659260">right.</span> <span m="1659410">So</span> <span m="1659500">that</span> <span m="1659620">gives</span> <span m="1659790">you an</span> <span m="1659860">idea.</span> <span m="1660120">You</span> <span m="1660380">have to do</span> <span m="1660550">a</span> <span m="1660590">lot</span> <span m="1660770">of</span> <span m="1660850">work</span> <span m="1661050">to</span> <span m="1661130">get</span> <span m="1661280">seven.</span> </p>
<p><span m="1664020">And</span> <span m="1664200">in</span> <span m="1664280">fact</span> <span m="1665230">it's</span> <span m="1665420">hard,</span> <span m="1665690">because</span> <span m="1665960">the</span> <span m="1666040">bottom</span> <span m="1666460">block</span> <span m="1666720">would</span> <span m="1666810">be</span> <span m="1666910">hanging</span> <span m="1667280">out</span> <span m="1667450">one</span> <span m="1667970">two</span> <span m="1668170">millionth</span> <span m="1668610">of</span> <span m="1668680">a</span> <span m="1668770">block.</span> <span m="1669640">All right?</span> <span m="1670000">So you're</span> <span m="1670170">adding</span> <span m="1670420">up</span> <span m="1670510">a</span> <span m="1670560">lot</span> <span m="1670820">of</span> <span m="1670890">tiny</span> <span m="1671220">things</span> <span m="1671670">to</span> <span m="1671970">get</span> <span m="1672180">to</span> <span m="1672290">seven</span> <span m="1672650">out there.</span> </p>
<p></p>
<p><span m="1674460">Now</span> <span m="1674650">it</span> <span m="1674740">turns</span> <span m="1675090">out</span> <span m="1675220">that</span> <span m="1675330">the</span> <span m="1675420">harmonic</span> <span m="1675920">numbers</span> <span m="1676860">grow</span> <span m="1677260">to</span> <span m="1677440">infinity.</span> <span m="1678800">And</span> <span m="1678980">we'll</span> <span m="1679080">see</span> <span m="1679280">that.</span> <span m="1679520">But</span> <span m="1679620">they</span> <span m="1679740">grow</span> <span m="1680020">very,</span> <span m="1680610">very</span> <span m="1680850">slowly.</span> <span m="1682760">And</span> <span m="1682930">to</span> <span m="1683000">see</span> <span m="1683240">that,</span> <span m="1683420">we</span> <span m="1683520">need</span> <span m="1683670">to</span> <span m="1683740">get</span> <span m="1683890">a</span> <span m="1683930">closed-form</span> <span m="1684680">expression</span> <span m="1685890">for</span> <span m="1686020">this</span> <span m="1686200">sum.</span> </p>
<p><span m="1688130">Now</span> <span m="1688480">does</span> <span m="1688640">anybody</span> <span m="1689030">know</span> <span m="1689850">a</span> <span m="1689940">closed-form</span> <span m="1690570">expression</span> <span m="1691210">for</span> <span m="1691340">the</span> <span m="1691420">sum</span> <span m="1691700">of</span> <span m="1691810">1</span> <span m="1691980">over</span> <span m="1692220">i?</span> <span m="1694690">I</span> <span m="1695130">don't</span> <span m="1695330">think</span> <span m="1695620">so,</span> <span m="1696040">because</span> <span m="1696200">I</span> <span m="1696570">don't</span> <span m="1696750">know</span> <span m="1697560">anybody</span> <span m="1697960">who</span> <span m="1698120">knows</span> <span m="1698710">that.</span> <span m="1700575">In</span> <span m="1701020">fact</span> <span m="1701290">it's</span> <span m="1701410">[? probably ?]</span> <span m="1701710">believed</span> <span m="1702220">that it</span> <span m="1702350">doesn't</span> <span m="1702620">exist.</span> </p>
<p><span m="1705150">So</span> <span m="1705520">what</span> <span m="1705790">do</span> <span m="1705860">we</span> <span m="1706000">do</span> <span m="1706440">to</span> <span m="1706580">get</span> <span m="1707710">a</span> <span m="1707810">good</span> <span m="1708280">estimation</span> <span m="1709360">of</span> <span m="1709540">that</span> <span m="1709760">sum?</span> <span m="1714360">What</span> <span m="1714530">do</span> <span m="1714590">we</span> <span m="1714670">do?</span> </p>
<p><span m="1716110">AUDIENCE: Integration bounds.</span> </p>
<p><span m="1717060">PROFESSOR: The integration</span> <span m="1717720">bounds.</span> <span m="1718110">Yeah.</span> <span m="1718450">Because</span> <span m="1718550">we're not</span> <span m="1718780">going</span> <span m="1718910">to</span> <span m="1718970">get</span> <span m="1719100">a</span> <span m="1719130">closed</span> <span m="1719550">form.</span> <span m="1720410">At</span> <span m="1720520">least</span> <span m="1721430">I</span> <span m="1721530">don't</span> <span m="1721660">know</span> <span m="1721770">of</span> <span m="1721870">anybody</span> <span m="1722200">who</span> <span m="1722290">knows</span> <span m="1722440">how</span> <span m="1722540">to</span> <span m="1722620">do</span> <span m="1722860">it.</span> <span m="1724260">So</span> <span m="1724430">we'll</span> <span m="1724530">use</span> <span m="1724720">the</span> <span m="1724800">integration</span> <span m="1725330">bounds.</span> </p>
<p><span m="1737260">And</span> <span m="1737540">what</span> <span m="1737880">kind</span> <span m="1738090">of</span> <span m="1738170">function</span> <span m="1738730">are</span> <span m="1738870">we</span> <span m="1739010">summing?</span> <span m="1739460">Is</span> <span m="1739630">it</span> <span m="1739880">increasing</span> <span m="1740420">or</span> <span m="1740480">decreasing?</span> <span m="1743000">Decreasing.</span> <span m="1743660">Yep.</span> </p>
<p><span m="1745790">So</span> <span m="1745950">we'll</span> <span m="1746070">use</span> <span m="1746280">the</span> <span m="1746340">formula</span> <span m="1746970">we</span> <span m="1747140">did</span> <span m="1747330">last</span> <span m="1747660">time.</span> <span m="1754380">All</span> <span m="1754570">right.</span> <span m="1754720">And</span> <span m="1754810">that</span> <span m="1754970">formula</span> <span m="1755910">is</span> <span m="1756380">that</span> <span m="1756510">the</span> <span m="1756600">sum</span> <span m="1757870">i</span> <span m="1758110">equals</span> <span m="1758390">1</span> <span m="1758770">to</span> <span m="1759470">n</span> <span m="1761550">of</span> <span m="1762582">f of</span> <span m="1763048">i</span> <span m="1763980">is,</span> <span m="1764370">at</span> <span m="1764570">most,</span> <span m="1765700">the</span> <span m="1765800">first</span> <span m="1766030">term</span> <span m="1767150">plus</span> <span m="1767470">the</span> <span m="1767620">integral.</span> <span m="1771730">And</span> <span m="1771950">it's</span> <span m="1772130">at</span> <span m="1772250">least</span> <span m="1773500">the</span> <span m="1773590">last</span> <span m="1773990">term</span> <span m="1774760">plus</span> <span m="1775090">the</span> <span m="1775210">integral.</span> <span m="1782220">All</span> <span m="1782440">right.</span> <span m="1782600">In</span> <span m="1782700">this</span> <span m="1782890">case</span> <span m="1783250">we</span> <span m="1783370">have</span> <span m="1784275">f of i</span> <span m="1784660">as 1</span> <span m="1784850">over</span> <span m="1785265">i.</span> </p>
<p><span m="1791140">So</span> <span m="1791380">the</span> <span m="1791460">first</span> <span m="1791770">step</span> <span m="1792950">is</span> <span m="1793190">to</span> <span m="1793270">compute</span> <span m="1793630">the</span> <span m="1793720">integral.</span> <span m="1802250">And</span> <span m="1802380">that's</span> <span m="1802590">pretty</span> <span m="1802820">easy.</span> <span m="1803530">That's</span> <span m="1803720">just</span> <span m="1803960">the</span> <span m="1804030">natural</span> <span m="1804440">log</span> <span m="1804720">of</span> <span m="1804840">x</span> <span m="1806890">evaluated</span> <span m="1807260">at</span> <span m="1807490">n</span> <span m="1807740">and</span> <span m="1808060">1.</span> <span m="1808520">And</span> <span m="1808660">that's</span> <span m="1808910">easy.</span> <span m="1810070">That's</span> <span m="1810200">just</span> <span m="1810440">the</span> <span m="1810510">natural</span> <span m="1810900">log</span> <span m="1811110">of</span> <span m="1811180">n.</span> <span m="1812990">So</span> <span m="1813140">the</span> <span m="1813280">integral</span> <span m="1813700">is</span> <span m="1813800">very</span> <span m="1814150">easy</span> <span m="1814450">here.</span> </p>
<p><span m="1815930">And</span> <span m="1816120">now</span> <span m="1816330">we</span> <span m="1816480">actually</span> <span m="1816830">can</span> <span m="1817000">just</span> <span m="1817200">plug</span> <span m="1817460">in</span> <span m="1817720">and</span> <span m="1817830">get</span> <span m="1818010">really</span> <span m="1818300">good</span> <span m="1818480">bounds.</span> <span m="1822100">Maybe</span> <span m="1822280">I'll</span> <span m="1822390">save</span> <span m="1822660">that</span> <span m="1822920">just</span> <span m="1823350">in</span> <span m="1823420">case.</span> <span m="1842600">OK</span> <span m="1842930">so</span> <span m="1843140">plugging</span> <span m="1843600">in</span> <span m="1844390">into</span> <span m="1844620">the</span> <span m="1844720">bounds</span> <span m="1845410">f</span> <span m="1845630">of</span> <span m="1845730">n</span> <span m="1846740">is</span> <span m="1846960">1</span> <span m="1847280">over</span> <span m="1847710">n</span> <span m="1849000">plus</span> <span m="1849390">the</span> <span m="1849530">integral</span> <span m="1849785">is</span> <span m="1850040">log</span> <span m="1850330">of</span> <span m="1850410">n</span> <span m="1851620">is</span> <span m="1851920">less</span> <span m="1852130">than</span> <span m="1852310">or</span> <span m="1852410">equal</span> <span m="1852670">to</span> <span m="1853550">the</span> <span m="1853670">n'th</span> <span m="1853840">harmonic</span> <span m="1854290">number.</span> <span m="1854860">And</span> <span m="1855310">it's</span> <span m="1855470">upper</span> <span m="1855720">bounded</span> <span m="1856120">by</span> <span m="1856310">f</span> <span m="1856590">of</span> <span m="1856700">1,</span> <span m="1857190">which</span> <span m="1857390">is</span> <span m="1857500">1</span> <span m="1859060">plus</span> <span m="1859810">the</span> <span m="1859950">integral.</span> </p>
<p><span m="1861798">All</span> <span m="1862260">right.</span> <span m="1862550">So</span> <span m="1862640">that's</span> <span m="1862920">really</span> <span m="1863310">good.</span> <span m="1863520">We</span> <span m="1863590">got</span> <span m="1863760">very</span> <span m="1864100">tight</span> <span m="1864390">bounds.</span> <span m="1865410">And</span> <span m="1865580">now</span> <span m="1865740">you</span> <span m="1865890">can</span> <span m="1866050">see</span> <span m="1866400">that</span> <span m="1866540">the</span> <span m="1866650">n'th</span> <span m="1866850">harmonic</span> <span m="1867280">number--</span> <span m="1868220">how</span> <span m="1868430">fast</span> <span m="1868800">is</span> <span m="1868930">it</span> <span m="1869020">growing?</span> <span m="1869800">If</span> <span m="1869880">I</span> <span m="1869960">were</span> <span m="1870050">to</span> <span m="1870110">use</span> <span m="1870300">tilde</span> <span m="1870580">notation,</span> <span m="1871100">what</span> <span m="1871360">would I</span> <span m="1871800">write?</span> <span m="1874620">As</span> <span m="1874830">log</span> <span m="1875030">n.</span> <span m="1875550">Yeah.</span> <span m="1875880">So</span> <span m="1876050">this</span> <span m="1876280">means</span> <span m="1876580">that</span> <span m="1877120">h of</span> <span m="1877350">n</span> <span m="1877730">is</span> <span m="1877880">tilde</span> <span m="1879210">log of</span> <span m="1879580">n.</span> </p>
<p><span m="1882750">Now</span> <span m="1883600">this</span> <span m="1883850">is</span> <span m="1883940">such</span> <span m="1884160">an</span> <span m="1884220">important</span> <span m="1884650">function</span> <span m="1885080">that</span> <span m="1885200">people</span> <span m="1885500">have</span> <span m="1885600">gone</span> <span m="1885950">to</span> <span m="1886090">a</span> <span m="1886160">lot</span> <span m="1886490">of</span> <span m="1886600">extra</span> <span m="1886890">work--</span> <span m="1887640">which</span> <span m="1887920">we</span> <span m="1888090">can't</span> <span m="1888430">do</span> <span m="1888630">in</span> <span m="1888710">this</span> <span m="1888870">class--</span> <span m="1890590">to</span> <span m="1891430">really</span> <span m="1891950">nail</span> <span m="1892240">down</span> <span m="1892810">the</span> <span m="1892950">value--</span> <span m="1893330">get it</span> <span m="1893550">much</span> <span m="1893770">closer.</span> <span m="1894260">And</span> <span m="1894380">in</span> <span m="1894460">fact</span> <span m="1896840">it's</span> <span m="1897280">now</span> <span m="1897640">known</span> <span m="1898090">that</span> <span m="1898230">it's</span> <span m="1898990">the</span> <span m="1899110">natural</span> <span m="1899530">log</span> <span m="1899750">of</span> <span m="1900050">n</span> <span m="1900560">plus</span> <span m="1901220">1</span> <span m="1901420">over</span> <span m="1901620">2n</span> <span m="1903420">plus</span> <span m="1903780">1</span> <span m="1903990">over</span> <span m="1904250">12n</span> <span m="1904940">squared</span> <span m="1906730">plus</span> <span m="1907700">epsilon</span> <span m="1908260">of</span> <span m="1908360">n</span> <span m="1909360">over</span> <span m="1909820">120n</span> <span m="1911390">to</span> <span m="1911480">the</span> <span m="1911550">fourth</span> <span m="1913030">where</span> <span m="1916360">for</span> <span m="1916650">all</span> <span m="1916920">n</span> <span m="1918080">epsilon</span> <span m="1918400">n</span> <span m="1918720">is</span> <span m="1918810">between</span> <span m="1919250">0</span> <span m="1920300">and</span> <span m="1920480">1.</span> </p>
<p><span m="1924820">Oop.</span> <span m="1925160">And I</span> <span m="1926160">left</span> <span m="1926460">one</span> <span m="1926670">piece</span> <span m="1926970">out.</span> <span m="1928230">Actually</span> <span m="1928570">there's</span> <span m="1928700">a</span> <span m="1928750">constant</span> <span m="1929300">term</span> <span m="1929580">in</span> <span m="1929670">here--</span> <span m="1931190">delta--</span> <span m="1931990">that's</span> <span m="1932200">called</span> <span m="1932540">Euler's</span> <span m="1932910">constant.</span> <span m="1935230">Delta</span> <span m="1937310">is</span> <span m="1938370">called</span> <span m="1938790">Euler's</span> <span m="1939170">constant,</span> <span m="1941630">and</span> <span m="1941910">equals</span> <span m="1944220">0.577215664</span> <span m="1950420">and</span> <span m="1950540">some</span> <span m="1950670">other</span> <span m="1950850">stuff.</span> <span m="1952290">And</span> <span m="1952330">in</span> <span m="1952410">fact</span> <span m="1952750">to</span> <span m="1952850">this</span> <span m="1952860">day</span> <span m="1953070">people</span> <span m="1953370">don't</span> <span m="1953630">know</span> <span m="1953840">if</span> <span m="1953940">this</span> <span m="1954140">constant</span> <span m="1954670">is</span> <span m="1955050">rational</span> <span m="1955870">or</span> <span m="1956070">irrational.</span> <span m="1957760">But</span> <span m="1957920">they</span> <span m="1958050">know</span> <span m="1958310">it</span> <span m="1958390">to</span> <span m="1958510">a</span> <span m="1958570">lot</span> <span m="1958840">of</span> <span m="1959070">decimal</span> <span m="1959420">places.</span> </p>
<p><span m="1962200">So</span> <span m="1963160">basically</span> <span m="1963650">H</span> <span m="1964130">n is</span> <span m="1964300">the</span> <span m="1964370">log</span> <span m="1964780">of</span> <span m="1965080">n</span> <span m="1965420">plus</span> <span m="1965850">some</span> <span m="1966780">fixed</span> <span m="1967150">value</span> <span m="1967550">between</span> <span m="1967940">0</span> <span m="1968250">and</span> <span m="1968400">1--</span> <span m="1968640">as</span> <span m="1968770">we</span> <span m="1968880">knew</span> <span m="1969070">it</span> <span m="1969150">had</span> <span m="1969350">to</span> <span m="1969420">be--</span> <span m="1970110">plus</span> <span m="1970450">even</span> <span m="1970730">some</span> <span m="1970900">tail</span> <span m="1971230">terms</span> <span m="1971580">here</span> <span m="1971910">that</span> <span m="1972120">get</span> <span m="1972320">very</span> <span m="1972580">small</span> <span m="1972960">as</span> <span m="1973070">n</span> <span m="1973230">gets</span> <span m="1973460">large.</span> </p>
<p><span m="1975770">All</span> <span m="1975940">right.</span> <span m="1976130">So</span> <span m="1976290">say</span> <span m="1976580">I</span> <span m="1976640">wanted</span> <span m="1977010">to</span> <span m="1977130">using</span> <span m="1978300">this</span> <span m="1978460">greedy</span> <span m="1978740">strategy.</span> <span m="1980070">How</span> <span m="1980300">many</span> <span m="1980480">blocks</span> <span m="1980970">I</span> <span m="1981060">would</span> <span m="1981240">need</span> <span m="1981640">to</span> <span m="1981760">get</span> <span m="1982610">a</span> <span m="1982730">block</span> <span m="1983370">100</span> <span m="1984110">blocks</span> <span m="1984540">out</span> <span m="1984750">over</span> <span m="1984920">the</span> <span m="1985010">end</span> <span m="1985120">of</span> <span m="1985180">the</span> <span m="1985240">table.</span> <span m="1986700">So</span> <span m="1986870">I</span> <span m="1986930">want</span> <span m="1987160">to</span> <span m="1987220">go</span> <span m="1987340">100</span> <span m="1987880">blocks</span> <span m="1988280">out.</span> <span m="1989300">How</span> <span m="1989470">many</span> <span m="1989620">blocks</span> <span m="1989950">do</span> <span m="1990070">I</span> <span m="1990160">need</span> <span m="1990680">using</span> <span m="1990990">this</span> <span m="1991120">strategy,</span> <span m="1993390">roughly?</span> <span m="1998090">What is</span> <span m="1998250">it?</span> <span m="2000390">I</span> <span m="2000520">think</span> <span m="2000620">somebody</span> <span m="2000680">said</span> <span m="2000930">it.</span> </p>
<p><span m="2003190">So</span> <span m="2003350">I</span> <span m="2003410">want</span> <span m="2003690">to</span> <span m="2003750">get</span> <span m="2005960">r</span> <span m="2006395">1</span> <span m="2006830">to</span> <span m="2006950">be</span> <span m="2007050">100.</span> <span m="2010020">So</span> <span m="2010500">I</span> <span m="2010610">need</span> <span m="2011470">the</span> <span m="2011590">harmonic</span> <span m="2012080">number</span> <span m="2012390">to</span> <span m="2012480">be</span> <span m="2013560">200.</span> <span m="2016070">And</span> <span m="2016150">what</span> <span m="2016310">does</span> <span m="2016430">this</span> <span m="2016620">tell</span> <span m="2016870">me</span> <span m="2017730">about</span> <span m="2018150">n</span> <span m="2018530">to</span> <span m="2018620">get</span> <span m="2019540">this</span> <span m="2019740">to</span> <span m="2019830">be</span> <span m="2019910">200?</span> <span m="2023840">e</span> <span m="2023875">to</span> <span m="2023910">the</span> <span m="2023980">200.</span> <span m="2025490">All right.</span> <span m="2025600">So</span> <span m="2025820">you're</span> <span m="2025910">going</span> <span m="2026030">to</span> <span m="2026090">need</span> <span m="2027120">an</span> <span m="2027260">exponential</span> <span m="2028030">in</span> <span m="2028120">200</span> <span m="2028580">number</span> <span m="2028800">of</span> <span m="2028870">blocks,</span> <span m="2029360">which</span> <span m="2029600">now</span> <span m="2029810">we're</span> <span m="2029920">past</span> <span m="2030320">the</span> <span m="2030380">stars,</span> <span m="2031350">all right,</span> <span m="2031560">to</span> <span m="2031640">get</span> <span m="2031790">that</span> <span m="2031970">many.</span> </p>
<p><span m="2034580">All right.</span> <span m="2034840">Now</span> <span m="2034980">we</span> <span m="2035110">won't</span> <span m="2035820">prove</span> <span m="2036170">it</span> <span m="2036240">in</span> <span m="2036340">class,</span> <span m="2037660">but</span> <span m="2037860">this</span> <span m="2038060">strategy--</span> <span m="2038770">the</span> <span m="2038920">TA</span> <span m="2039480">greedy</span> <span m="2039880">strategy--</span> <span m="2040780">is</span> <span m="2041290">optimal.</span> <span m="2042710">There</span> <span m="2042920">is</span> <span m="2043100">no</span> <span m="2043450">way</span> <span m="2044250">to</span> <span m="2044370">get</span> <span m="2044520">a</span> <span m="2044550">block</span> <span m="2044930">farther</span> <span m="2045300">out</span> <span m="2045440">over</span> <span m="2045610">the</span> <span m="2045750">edge.</span> </p>
<p><span m="2047660">That</span> <span m="2047950">said,</span> <span m="2048310">there</span> <span m="2048429">is</span> <span m="2048659">a</span> <span m="2048719">second</span> <span m="2049270">way</span> <span m="2049480">to</span> <span m="2049570">get</span> <span m="2049850">there.</span> <span m="2051080">And</span> <span m="2051230">in</span> <span m="2051290">the</span> <span m="2051360">second</span> <span m="2051810">way,</span> <span m="2051989">the</span> <span m="2052070">top</span> <span m="2052510">block</span> <span m="2052830">is</span> <span m="2053030">not</span> <span m="2054070">the</span> <span m="2054190">optimal</span> <span m="2054550">block.</span> <span m="2055380">In</span> <span m="2055510">the</span> <span m="2055580">second</span> <span m="2056020">way,</span> <span m="2056630">it's</span> <span m="2056870">the</span> <span m="2057050">second</span> <span m="2057460">block</span> <span m="2057830">that</span> <span m="2057920">goes</span> <span m="2058130">out.</span> <span m="2058389">So</span> <span m="2058489">it</span> <span m="2058590">looks</span> <span m="2058650">something</span> <span m="2058880">like</span> <span m="2059090">that.</span> <span m="2059529">So</span> <span m="2059969">there's</span> <span m="2060190">a</span> <span m="2060250">counterweight</span> <span m="2060810">on</span> <span m="2060949">it.</span> </p>
<p><span m="2061730">And</span> <span m="2061920">in</span> <span m="2062000">the</span> <span m="2062070">text,</span> <span m="2063150">when</span> <span m="2063300">you</span> <span m="2063389">read</span> <span m="2063550">chapter</span> <span m="2063880">nine,</span> <span m="2064320">you</span> <span m="2064429">will</span> <span m="2064530">see</span> <span m="2064750">the</span> <span m="2064860">proof</span> <span m="2065969">that</span> <span m="2066330">shows</span> <span m="2067000">that</span> <span m="2067389">there's</span> <span m="2067699">two</span> <span m="2067909">solutions</span> <span m="2068469">that</span> <span m="2068580">are</span> <span m="2068659">optimal.</span> <span m="2069000">One's</span> <span m="2069290">a</span> <span m="2069340">greedy,</span> <span m="2069670">and</span> <span m="2069790">one's this</span> <span m="2070199">other</span> <span m="2070420">one</span> <span m="2070639">that</span> <span m="2070880">sort</span> <span m="2071310">of</span> <span m="2071389">is</span> <span m="2071570">mostly</span> <span m="2072080">the</span> <span m="2072150">greedy,</span> <span m="2072400">except</span> <span m="2073060">the</span> <span m="2073179">top</span> <span m="2073469">couple of</span> <span m="2073730">blocks</span> <span m="2074080">are</span> <span m="2074130">different.</span> <span m="2074810">And</span> <span m="2074989">it</span> <span m="2075070">proves</span> <span m="2075420">that's</span> <span m="2075650">optimal.</span> <span m="2076150">We</span> <span m="2076699">won't</span> <span m="2076880">cover</span> <span m="2077080">that</span> <span m="2077239">in</span> <span m="2077320">class.</span> </p>
<p><span m="2080580">Now</span> <span m="2080929">actually</span> <span m="2081219">in</span> <span m="2081280">the</span> <span m="2081340">literature</span> <span m="2081780">today</span> <span m="2082570">people</span> <span m="2082830">have</span> <span m="2082920">been</span> <span m="2083030">studying</span> <span m="2084409">how</span> <span m="2084670">far</span> <span m="2084949">a</span> <span m="2085010">block</span> <span m="2085290">can</span> <span m="2085420">go</span> <span m="2085580">out</span> <span m="2085900">if</span> <span m="2086040">you're</span> <span m="2086150">allowed</span> <span m="2086510">to</span> <span m="2086580">do</span> <span m="2086750">more</span> <span m="2087090">than</span> <span m="2087260">one</span> <span m="2087480">block</span> <span m="2087770">per</span> <span m="2087900">level,</span> <span m="2088310">like</span> <span m="2088590">one</span> <span m="2088790">of</span> <span m="2088860">the</span> <span m="2089300">students</span> <span m="2089719">wanted</span> <span m="2089949">to</span> <span m="2090040">do.</span> <span m="2090870">So</span> <span m="2091110">if</span> <span m="2091210">you're</span> <span m="2091310">allowed</span> <span m="2091620">to</span> <span m="2091719">have</span> <span m="2092480">multiple</span> <span m="2093040">blocks</span> <span m="2093429">on</span> <span m="2093510">the</span> <span m="2093580">same</span> <span m="2093940">level,</span> <span m="2095510">it</span> <span m="2095560">turns</span> <span m="2095880">out</span> <span m="2096030">you</span> <span m="2096120">can</span> <span m="2096250">get</span> <span m="2096739">much</span> <span m="2097100">farther</span> <span m="2097460">out</span> <span m="2097670">with n</span> <span m="2097930">blocks.</span> <span m="2098900">And</span> <span m="2099190">right</span> <span m="2099400">now</span> <span m="2099630">it's</span> <span m="2099800">known</span> <span m="2100170">that</span> <span m="2100330">it's</span> <span m="2101300">something</span> <span m="2101800">like</span> <span m="2102100">n</span> <span m="2102400">to</span> <span m="2102480">the</span> <span m="2102560">third</span> <span m="2103250">is</span> <span m="2103440">how</span> <span m="2103590">far</span> <span m="2103830">you</span> <span m="2103920">can</span> <span m="2104060">get</span> <span m="2104200">out,</span> <span m="2104830">which</span> <span m="2105110">is</span> <span m="2105240">much</span> <span m="2105590">better</span> <span m="2105860">than</span> <span m="2106370">log</span> <span m="2106600">of</span> <span m="2106710">n.</span> <span m="2107310">And</span> <span m="2107430">so</span> <span m="2107670">mathematicians</span> <span m="2108370">have</span> <span m="2108470">been</span> <span m="2108640">studying</span> <span m="2109040">that</span> <span m="2109190">problem.</span> </p>
<p><span m="2110950">Any</span> <span m="2111140">questions</span> <span m="2113110">about</span> <span m="2114510">the</span> <span m="2114620">block</span> <span m="2114930">stacking?</span> <span m="2118360">OK.</span> <span m="2119430">All</span> <span m="2119600">right.</span> <span m="2119830">So</span> <span m="2121630">that's</span> <span m="2122460">basically</span> <span m="2123090">it</span> <span m="2123230">for</span> <span m="2123400">sums.</span> <span m="2124130">I</span> <span m="2124220">do</span> <span m="2124380">want</span> <span m="2124540">to</span> <span m="2124600">spend</span> <span m="2124780">a</span> <span m="2124810">little</span> <span m="2124990">time</span> <span m="2125230">talking</span> <span m="2125560">about</span> <span m="2125900">products.</span> <span m="2127190">And</span> <span m="2127350">that</span> <span m="2127510">may</span> <span m="2127680">get</span> <span m="2127840">us</span> <span m="2127960">back</span> <span m="2128200">into</span> <span m="2128350">sums</span> <span m="2128630">pretty</span> <span m="2128880">quick.</span> </p>
<p><span m="2132410">The</span> <span m="2132540">most</span> <span m="2132830">famous</span> <span m="2133290">product</span> <span m="2134450">out</span> <span m="2134740">there</span> <span m="2135200">is</span> <span m="2135580">n</span> <span m="2135810">factorial.</span> <span m="2137690">And</span> <span m="2138350">once</span> <span m="2138650">we</span> <span m="2138760">get</span> <span m="2139010">to</span> <span m="2139150">counting</span> <span m="2139415">and</span> <span m="2139740">probability,</span> <span m="2140900">probably</span> <span m="2141450">every</span> <span m="2141720">lecture</span> <span m="2142260">we're</span> <span m="2142280">going</span> <span m="2142410">to</span> <span m="2142470">be</span> <span m="2142670">looking</span> <span m="2142950">at</span> <span m="2143040">n</span> <span m="2143170">factorial</span> <span m="2143780">one</span> <span m="2143950">way</span> <span m="2144090">or</span> <span m="2144170">another.</span> <span m="2145530">Just</span> <span m="2145760">comes</span> <span m="2145980">up</span> <span m="2146120">all</span> <span m="2146300">the</span> <span m="2146390">time.</span> <span m="2148595">n</span> <span m="2149040">factorial</span> <span m="2150010">is</span> <span m="2150120">the</span> <span m="2150190">product</span> <span m="2152410">of</span> <span m="2152690">i</span> <span m="2152900">equals</span> <span m="2153020">1</span> <span m="2153350">to n</span> <span m="2153750">of i.</span> <span m="2154610">It's</span> <span m="2154830">the</span> <span m="2154910">product</span> <span m="2155530">of</span> <span m="2155600">the</span> <span m="2155680">first</span> <span m="2156230">n</span> <span m="2157560">natural</span> <span m="2157950">numbers</span> <span m="2159480">after</span> <span m="2159780">zero.</span> </p>
<p><span m="2162530">Now</span> <span m="2162770">we'd</span> <span m="2162950">like</span> <span m="2163180">to</span> <span m="2163270">be</span> <span m="2163380">able</span> <span m="2163630">to</span> <span m="2163870">get</span> <span m="2164040">a</span> <span m="2164070">closed-form</span> <span m="2164770">expression</span> <span m="2165280">for</span> <span m="2165400">this,</span> <span m="2165760">because</span> <span m="2165980">factorial</span> <span m="2166600">is</span> <span m="2166700">really</span> <span m="2166860">not</span> <span m="2167000">closed</span> <span m="2167310">form,</span> <span m="2167600">because</span> <span m="2167760">it's</span> <span m="2168440">hiding</span> <span m="2168860">this</span> <span m="2169070">thing.</span> <span m="2170870">And</span> <span m="2171010">if</span> <span m="2171080">I</span> <span m="2171250">ask</span> <span m="2171500">you</span> <span m="2171600">how</span> <span m="2171800">big</span> <span m="2172000">is</span> <span m="2172120">n</span> <span m="2172270">factorial,</span> <span m="2174370">sort</span> <span m="2174650">of</span> <span m="2174740">hard</span> <span m="2175040">to</span> <span m="2175470">say</span> <span m="2176030">how</span> <span m="2176210">big</span> <span m="2176430">it</span> <span m="2176510">is,</span> <span m="2177030">how</span> <span m="2177090">fast</span> <span m="2177420">it</span> <span m="2177490">grows.</span> <span m="2179560">Any</span> <span m="2179790">ideas</span> <span m="2180260">about</span> <span m="2180710">how</span> <span m="2180940">we</span> <span m="2181040">might</span> <span m="2181250">go</span> <span m="2181390">about</span> <span m="2181730">doing</span> <span m="2182440">a</span> <span m="2182560">product</span> <span m="2184790">to</span> <span m="2184850">get</span> <span m="2184970">a</span> <span m="2185000">closed-form</span> <span m="2185560">expression?</span> </p>
<p><span m="2187160">AUDIENCE: If you take</span> <span m="2187580">a</span> <span m="2187610">logarithm</span> <span m="2187640">of</span> <span m="2188066">a</span> <span m="2188492">product, you get</span> <span m="2188918">a sum.</span> </p>
<p><span m="2189770">PROFESSOR: Yeah,</span> <span m="2190390">good.</span> <span m="2191170">Take</span> <span m="2191420">the</span> <span m="2191490">log</span> <span m="2191830">of</span> <span m="2191910">it.</span> <span m="2196970">Well,</span> <span m="2199330">let's</span> <span m="2199560">write</span> <span m="2199780">that</span> <span m="2199940">out--</span> <span m="2200190">1</span> <span m="2200440">times</span> <span m="2200700">2</span> <span m="2200940">times</span> <span m="2201250">3</span> <span m="2202130">to</span> <span m="2202570">n.</span> <span m="2203980">That</span> <span m="2204220">equals</span> <span m="2204920">the</span> <span m="2205000">log</span> <span m="2205300">of</span> <span m="2205410">1</span> <span m="2206210">plus</span> <span m="2206490">the</span> <span m="2206570">log</span> <span m="2206870">of</span> <span m="2206980">2.</span> <span m="2210410">Whoops.</span> <span m="2213600">All</span> <span m="2213780">right.</span> <span m="2213940">So</span> <span m="2214050">I've</span> <span m="2214150">now</span> <span m="2214430">got</span> <span m="2214850">a</span> <span m="2214920">sum.</span> </p>
<p><span m="2221005">And</span> <span m="2221480">now</span> <span m="2221630">that</span> <span m="2221740">I've</span> <span m="2221780">got</span> <span m="2221940">a</span> <span m="2222020">sum</span> <span m="2223290">we</span> <span m="2223440">can</span> <span m="2223580">use</span> <span m="2223780">all</span> <span m="2224060">the</span> <span m="2224390">tools</span> <span m="2225150">we</span> <span m="2225280">had</span> <span m="2225460">for</span> <span m="2225570">sums.</span> <span m="2226650">So</span> <span m="2226830">really,</span> <span m="2227200">products</span> <span m="2227790">look</span> <span m="2228010">hairy,</span> <span m="2229460">but</span> <span m="2229630">they're</span> <span m="2229750">no</span> <span m="2230010">different</span> <span m="2230340">than</span> <span m="2230440">sums.</span> <span m="2231430">Because</span> <span m="2231620">we're</span> <span m="2231710">going</span> <span m="2231850">to</span> <span m="2231970">get</span> <span m="2232290">an</span> <span m="2232380">approximation</span> <span m="2233190">or</span> <span m="2233370">get</span> <span m="2233520">an</span> <span m="2233610">answer</span> <span m="2233920">for</span> <span m="2234520">[? login ?]</span> <span m="2234795">n</span> <span m="2235070">factorial.</span> <span m="2235640">Then</span> <span m="2235680">we'll</span> <span m="2235830">exponentiate</span> <span m="2236205">it, and</span> <span m="2236580">we'll</span> <span m="2236710">have</span> <span m="2236860">the</span> <span m="2236970">answer.</span> </p>
<p><span m="2238780">All</span> <span m="2238970">right.</span> <span m="2239160">So</span> <span m="2239280">we're</span> <span m="2239370">looking</span> <span m="2239540">at</span> <span m="2239680">this</span> <span m="2240145">sum.</span> <span m="2241540">And</span> <span m="2242500">so</span> <span m="2242680">how</span> <span m="2242840">do</span> <span m="2242940">I</span> <span m="2243000">get</span> <span m="2243140">a</span> <span m="2243170">closed</span> <span m="2243600">form</span> <span m="2243940">for</span> <span m="2244070">that?</span> <span m="2246200">Turns</span> <span m="2246560">out</span> <span m="2247130">nobody</span> <span m="2247510">knows</span> <span m="2248260">the</span> <span m="2248350">closed</span> <span m="2248700">form.</span> <span m="2250010">But</span> <span m="2250210">we</span> <span m="2250320">do</span> <span m="2250560">know</span> <span m="2250750">a</span> <span m="2250810">method</span> <span m="2251530">that</span> <span m="2251660">approximates</span> <span m="2252320">it</span> <span m="2252410">very</span> <span m="2252640">well,</span> <span m="2255100">which</span> <span m="2255390">is</span> <span m="2256770">the</span> <span m="2256900">integration</span> <span m="2257450">bounce.</span> <span m="2259470">Only</span> <span m="2259770">in</span> <span m="2259860">this</span> <span m="2260040">case,</span> <span m="2260700">it's</span> <span m="2260790">an</span> <span m="2260870">increasing</span> <span m="2261390">function.</span> </p>
<p><span m="2263450">So</span> <span m="2263520">let's</span> <span m="2263710">do</span> <span m="2263850">that.</span> <span m="2285450">OK.</span> <span m="2285960">So</span> <span m="2286240">let's</span> <span m="2286420">do</span> <span m="2286540">that.</span> <span m="2301620">And</span> <span m="2301760">the</span> <span m="2301820">formula</span> <span m="2302330">is</span> <span m="2302440">easy.</span> <span m="2302760">It's</span> <span m="2302960">basically</span> <span m="2303410">the</span> <span m="2303510">same</span> <span m="2303790">formula</span> <span m="2304150">we</span> <span m="2304270">had</span> <span m="2304490">there--</span> <span m="2305770">that</span> <span m="2305930">the</span> <span m="2306000">sum</span> <span m="2308020">i</span> <span m="2308240">equals</span> <span m="2308430">1</span> <span m="2308620">to</span> <span m="2308710">n</span> <span m="2309020">of f of</span> <span m="2309270">i</span> <span m="2310740">is</span> <span m="2310980">at</span> <span m="2311080">most</span> <span m="2312530">the</span> <span m="2312580">n'th</span> <span m="2312870">term</span> <span m="2314200">plus</span> <span m="2314420">the</span> <span m="2314530">integral.</span> <span m="2317950">And</span> <span m="2318190">it's</span> <span m="2318330">at</span> <span m="2318430">least</span> <span m="2319420">the</span> <span m="2319570">first</span> <span m="2319920">term</span> <span m="2321770">plus</span> <span m="2322040">the</span> <span m="2322140">integral.</span> </p>
<p><span m="2327006">All</span> <span m="2327410">right.</span> <span m="2327760">Now</span> <span m="2327910">in</span> <span m="2327990">this</span> <span m="2328160">case</span> <span m="2329150">f of</span> <span m="2329430">i</span> <span m="2329920">is</span> <span m="2330150">the</span> <span m="2330240">log</span> <span m="2330590">of</span> <span m="2330670">i.</span> <span m="2334890">And</span> <span m="2335120">so</span> <span m="2335240">we</span> <span m="2335320">need</span> <span m="2335540">to</span> <span m="2335610">compute</span> <span m="2336010">the</span> <span m="2336140">integral</span> <span m="2339840">of</span> <span m="2340160">log</span> <span m="2340460">of</span> <span m="2340590">x.</span> <span m="2343300">And</span> <span m="2343410">that's</span> <span m="2343650">easy.</span> <span m="2344060">That's</span> <span m="2344320">x</span> <span m="2344780">ln</span> <span m="2345240">of</span> <span m="2345700">x</span> <span m="2346500">minus</span> <span m="2347010">x</span> <span m="2348290">evaluated</span> <span m="2349400">at n</span> <span m="2349900">and 1.</span> <span m="2351950">And</span> <span m="2352270">that's</span> <span m="2352470">just</span> <span m="2352900">n</span> <span m="2354000">log</span> <span m="2354490">of</span> <span m="2354600">n</span> <span m="2354760">minus</span> <span m="2355170">n.</span> <span m="2359410">If</span> <span m="2359480">I</span> <span m="2359540">plug in</span> <span m="2360040">1,</span> <span m="2360580">I</span> <span m="2360680">get</span> <span m="2360690">a</span> <span m="2360740">0</span> <span m="2361180">here,</span> <span m="2361950">and</span> <span m="2362050">I'm</span> <span m="2362120">subtracting</span> <span m="2362690">negative</span> <span m="2362970">1</span> <span m="2363250">there.</span> </p>
<p><span m="2366070">All</span> <span m="2366220">right.</span> <span m="2366400">And</span> <span m="2366520">so</span> <span m="2366600">now</span> <span m="2366940">we'll</span> <span m="2366970">just</span> <span m="2367240">plug</span> <span m="2368750">this</span> <span m="2369320">into</span> <span m="2369580">the</span> <span m="2369710">integral</span> <span m="2370720">and</span> <span m="2370910">we'll</span> <span m="2371010">get</span> <span m="2371170">our</span> <span m="2371270">bounce.</span> <span m="2374890">So</span> <span m="2375420">f</span> <span m="2375720">of</span> <span m="2375850">1.</span> <span m="2377700">What</span> <span m="2377850">is</span> <span m="2378030">f</span> <span m="2378220">of</span> <span m="2378330">1</span> <span m="2380000">in</span> <span m="2380080">this</span> <span m="2380230">case?</span> <span m="2382370">Yeah,</span> <span m="2382640">0.</span> <span m="2384980">Plus</span> <span m="2386740">n</span> <span m="2387440">ln</span> <span m="2388846">n</span> <span m="2389230">minus</span> <span m="2389700">n</span> <span m="2390050">plus</span> <span m="2390370">1</span> <span m="2391770">less</span> <span m="2392010">than</span> <span m="2392140">equal</span> <span m="2392420">to</span> <span m="2394300">log</span> <span m="2394690">of</span> <span m="2394800">n</span> <span m="2394930">factorial,</span> <span m="2397280">less</span> <span m="2397500">than</span> <span m="2397630">or</span> <span m="2397720">equal</span> <span m="2397930">to</span> <span m="2398180">f</span> <span m="2398410">of</span> <span m="2398510">n</span> <span m="2399580">is</span> <span m="2400050">log of</span> <span m="2400360">n.</span> </p>
<p><span m="2407830">OK.</span> <span m="2409340">So</span> <span m="2409600">to</span> <span m="2409680">get</span> <span m="2409860">bounds</span> <span m="2410180">on</span> <span m="2410300">n</span> <span m="2410420">factorial,</span> <span m="2411100">we</span> <span m="2411200">just</span> <span m="2411690">exponentiate</span> <span m="2412590">both</span> <span m="2412800">sides.</span> <span m="2414770">So</span> <span m="2415040">I</span> <span m="2415120">take</span> <span m="2415540">e to</span> <span m="2415900">this,</span> <span m="2416310">gives</span> <span m="2416640">me--</span> <span m="2417920">well,</span> <span m="2418100">that's</span> <span m="2418300">going</span> <span m="2418420">to</span> <span m="2418500">give</span> <span m="2420640">n</span> <span m="2420940">to</span> <span m="2421030">the</span> <span m="2421200">n</span> <span m="2421560">here.</span> <span m="2423220">And</span> <span m="2423360">this</span> <span m="2423530">is</span> <span m="2423660">negative.</span> <span m="2424080">That'll</span> <span m="2424290">be</span> <span m="2424430">e to</span> <span m="2424770">the</span> <span m="2424870">n</span> <span m="2425010">minus</span> <span m="2425350">1.</span> <span m="2429690">And</span> <span m="2430010">here</span> <span m="2430390">I've</span> <span m="2430480">got</span> <span m="2432480">n</span> <span m="2432700">plus</span> <span m="2432960">1</span> <span m="2433250">times</span> <span m="2433450">the</span> <span m="2433550">log</span> <span m="2433800">of n.</span> <span m="2434170">So</span> <span m="2434410">it's n</span> <span m="2434700">to</span> <span m="2434780">the</span> <span m="2434900">n</span> <span m="2435030">plus</span> <span m="2435310">1</span> <span m="2436710">over</span> <span m="2437460">the</span> <span m="2437560">same</span> <span m="2437830">thing--</span> <span m="2438680">e</span> <span m="2438870">to</span> <span m="2438960">the</span> <span m="2439080">n</span> <span m="2439200">minus</span> <span m="2439520">1.</span> </p>
<p><span m="2441710">So</span> <span m="2441890">we</span> <span m="2441970">got</span> <span m="2442140">very</span> <span m="2442420">close</span> <span m="2442760">bounds.</span> <span m="2443160">They're</span> <span m="2443220">within</span> <span m="2443470">a</span> <span m="2443520">factor</span> <span m="2443940">of</span> <span m="2444050">n--</span> <span m="2445500">which</span> <span m="2445720">sounds</span> <span m="2446040">like</span> <span m="2446140">a</span> <span m="2446190">lot,</span> <span m="2446570">but</span> <span m="2446710">compared</span> <span m="2447050">to</span> <span m="2447120">how</span> <span m="2447240">big</span> <span m="2447470">n</span> <span m="2447630">factorial</span> <span m="2448320">is,</span> <span m="2448740">not</span> <span m="2449000">so</span> <span m="2449150">bad.</span> <span m="2450450">Pretty</span> <span m="2450980">good</span> <span m="2451160">bounds</span> <span m="2451490">there.</span> <span m="2454410">OK.</span> <span m="2457470">So</span> <span m="2457650">n</span> <span m="2457810">factorial's</span> <span m="2458370">about</span> <span m="2458920">n</span> <span m="2459130">over</span> <span m="2459350">e</span> <span m="2459650">to</span> <span m="2459720">the</span> <span m="2459850">n'th</span> <span m="2460060">power.</span> </p>
<p><span m="2461570">Now</span> <span m="2462880">this</span> <span m="2463130">is</span> <span m="2463220">so</span> <span m="2463380">important</span> <span m="2463910">that--</span> <span m="2464500">even</span> <span m="2464760">more</span> <span m="2464950">important than</span> <span m="2465410">harmonic</span> <span m="2465880">numbers--</span> <span m="2466640">people</span> <span m="2466920">have</span> <span m="2467010">gone</span> <span m="2467210">to</span> <span m="2467280">a</span> <span m="2467350">lot</span> <span m="2467670">of</span> <span m="2467770">effort</span> <span m="2468060">to</span> <span m="2468120">get</span> <span m="2468280">very</span> <span m="2468700">tight</span> <span m="2469020">bounds</span> <span m="2469460">on</span> <span m="2469600">n</span> <span m="2469710">factorial.</span> <span m="2471020">So</span> <span m="2471050">let</span> <span m="2471160">me</span> <span m="2471260">tell</span> <span m="2471450">you</span> <span m="2471540">what</span> <span m="2471660">those</span> <span m="2471910">are.</span> </p>
<p><span m="2479760">It's</span> <span m="2479880">called</span> <span m="2480190">Stirling's</span> <span m="2480770">formula.</span> <span m="2490510">And</span> <span m="2490640">that</span> <span m="2490800">says</span> <span m="2491120">that</span> <span m="2491250">n</span> <span m="2491430">factorial</span> <span m="2493050">equals</span> <span m="2494550">n</span> <span m="2494890">over</span> <span m="2495190">e</span> <span m="2495650">to</span> <span m="2495770">the</span> <span m="2495900">n'th</span> <span m="2496100">power</span> <span m="2497430">times</span> <span m="2498530">the</span> <span m="2498640">square</span> <span m="2498930">root</span> <span m="2499070">of</span> <span m="2499190">2</span> <span m="2499420">pi</span> <span m="2499810">n</span> <span m="2501530">times</span> <span m="2502230">e</span> <span m="2503420">to</span> <span m="2503570">the</span> <span m="2503730">epsilon</span> <span m="2504134">n</span> <span m="2506110">where</span> <span m="2509860">epsilon</span> <span m="2510590">n</span> <span m="2510990">is</span> <span m="2511160">between</span> <span m="2511750">1</span> <span m="2512290">over</span> <span m="2513710">12n</span> <span m="2514770">plus</span> <span m="2515110">1</span> <span m="2518160">and</span> <span m="2518550">1</span> <span m="2518700">over</span> <span m="2518850">12n.</span> <span m="2520670">So</span> <span m="2520930">it's</span> <span m="2521170">epsilon</span> <span m="2521600">n</span> <span m="2521740">is</span> <span m="2521860">going</span> <span m="2522210">to</span> <span m="2522290">0</span> <span m="2522680">as</span> <span m="2522800">n</span> <span m="2522970">gets</span> <span m="2523170">big.</span> <span m="2524430">So</span> <span m="2524960">e to</span> <span m="2525280">something</span> <span m="2525550">going</span> <span m="2525740">to</span> <span m="2525810">0</span> <span m="2526120">is</span> <span m="2526220">going</span> <span m="2526510">to</span> <span m="2526620">1.</span> </p>
<p><span m="2528020">All</span> <span m="2528200">right.</span> <span m="2528420">So</span> <span m="2528680">another</span> <span m="2528950">way</span> <span m="2529220">of</span> <span m="2530410">looking</span> <span m="2530720">at</span> <span m="2530800">this</span> <span m="2531000">is</span> <span m="2531100">you</span> <span m="2531170">could</span> <span m="2531300">put</span> <span m="2531430">a tilde</span> <span m="2531840">here,</span> <span m="2533060">all right,</span> <span m="2533270">and</span> <span m="2533460">ignore</span> <span m="2533860">that</span> <span m="2534080">term.</span> <span m="2534490">And that's</span> <span m="2534800">called</span> <span m="2535030">Stirling's</span> <span m="2535520">formula.</span> <span m="2536310">This</span> <span m="2536540">is</span> <span m="2536660">a</span> <span m="2536720">good</span> <span m="2537000">thing</span> <span m="2537800">to</span> <span m="2537920">have</span> <span m="2538070">on</span> <span m="2538140">your</span> <span m="2538250">crib</span> <span m="2538470">sheet</span> <span m="2539670">for</span> <span m="2539810">the</span> <span m="2539890">midterm.</span> <span m="2541110">So</span> <span m="2541240">very</span> <span m="2541670">likely</span> <span m="2542320">there'll</span> <span m="2542480">be</span> <span m="2542610">something</span> <span m="2543040">where</span> <span m="2543140">you've</span> <span m="2543210">got</span> <span m="2543390">to</span> <span m="2543510">have</span> <span m="2543860">some</span> <span m="2544200">kind</span> <span m="2544440">of</span> <span m="2545030">analysis</span> <span m="2545700">of</span> <span m="2545810">n</span> <span m="2545960">factorial,</span> <span m="2547010">and</span> <span m="2547150">to</span> <span m="2547210">know</span> <span m="2547570">how</span> <span m="2547740">fast</span> <span m="2548230">it's</span> <span m="2548400">growing.</span> </p>
<p><span m="2551370">Now</span> <span m="2552090">these</span> <span m="2552310">bounds</span> <span m="2552750">you</span> <span m="2552840">get</span> <span m="2553270">here</span> <span m="2553610">are</span> <span m="2553710">very</span> <span m="2554050">tight.</span> <span m="2554860">For</span> <span m="2554980">example,</span> <span m="2556720">100</span> <span m="2557220">factorial</span> <span m="2559290">is</span> <span m="2559550">at</span> <span m="2559690">least</span> <span m="2561160">100</span> <span m="2562330">over</span> <span m="2562670">e</span> <span m="2563780">to</span> <span m="2563930">100</span> <span m="2565400">times</span> <span m="2565870">square</span> <span m="2566150">root</span> <span m="2566410">200</span> <span m="2566920">pi,</span> <span m="2569030">times</span> <span m="2570040">e</span> <span m="2570140">to the</span> <span m="2570230">1</span> <span m="2570520">over</span> <span m="2570800">1,201.</span> <span m="2572960">This</span> <span m="2573190">thing</span> <span m="2573390">is</span> <span m="2573810">just</span> <span m="2574220">really</span> <span m="2574450">tiny.</span> <span m="2574880">This</span> <span m="2575120">thing</span> <span m="2575390">here</span> <span m="2576590">is</span> <span m="2577210">1.0008329</span> <span m="2581370">and</span> <span m="2581490">something.</span> <span m="2583080">And</span> <span m="2583250">100</span> <span m="2583590">factorial</span> <span m="2584240">is</span> <span m="2584360">lower-bounded</span> <span m="2586710">by</span> <span m="2586890">the</span> <span m="2586980">same</span> <span m="2587280">things,</span> <span m="2589880">only</span> <span m="2590150">here</span> <span m="2590510">it's</span> <span m="2590730">e</span> <span m="2591010">to the</span> <span m="2591110">1</span> <span m="2591350">over</span> <span m="2591510">1,200.</span> <span m="2595040">And</span> <span m="2595330">that</span> <span m="2595650">is</span> <span m="2596110">very</span> <span m="2596970">close</span> <span m="2597220">to</span> <span m="2597310">1--</span> <span m="2598490">1.00083368.</span> </p>
<p><span m="2604100">So</span> <span m="2604280">the</span> <span m="2604400">difference</span> <span m="2605000">in</span> <span m="2605090">these</span> <span m="2605320">bounds</span> <span m="2606850">is</span> <span m="2607050">a</span> <span m="2607330">tiny</span> <span m="2607960">fraction</span> <span m="2608440">of</span> <span m="2608530">1%.</span> <span m="2610770">So</span> <span m="2610990">the</span> <span m="2611090">bound</span> <span m="2611540">you</span> <span m="2611650">get</span> <span m="2611870">from</span> <span m="2612020">Stirling's</span> <span m="2612490">formula</span> <span m="2613360">is</span> <span m="2613570">very,</span> <span m="2614060">very</span> <span m="2614200">close</span> <span m="2614610">to</span> <span m="2614690">the</span> <span m="2614780">right</span> <span m="2615000">answer.</span> </p>
<p><span m="2621910">Any</span> <span m="2622100">questions</span> <span m="2624770">on</span> <span m="2624930">Stirling's</span> <span m="2625320">formula?</span> <span m="2626670">We</span> <span m="2626810">won't</span> <span m="2627030">derive</span> <span m="2627370">it.</span> <span m="2627450">You</span> <span m="2627530">can</span> <span m="2627650">derive</span> <span m="2628000">that</span> <span m="2628220">in</span> <span m="2628320">a</span> <span m="2628380">graduate</span> <span m="2629030">math</span> <span m="2629290">class.</span> <span m="2629620">They</span> <span m="2629730">do</span> <span m="2629850">that</span> <span m="2630050">sometimes.</span> </p>
<p><span m="2633380">More</span> <span m="2633580">typically</span> <span m="2634060">you'll</span> <span m="2634190">see</span> <span m="2634440">it</span> <span m="2634550">without</span> <span m="2634900">that</span> <span m="2635250">e</span> <span m="2635615">to</span> <span m="2635980">the</span> <span m="2636150">error</span> <span m="2636450">term.</span> <span m="2636940">And</span> <span m="2637040">you</span> <span m="2637120">just</span> <span m="2637340">see</span> <span m="2637490">n</span> <span m="2637690">factorial</span> <span m="2638300">written</span> <span m="2638540">as</span> <span m="2638690">tilde,</span> <span m="2640170">and</span> <span m="2640400">over</span> <span m="2641120">e</span> <span m="2641620">to</span> <span m="2641790">the</span> <span m="2641950">n,</span> <span m="2643070">square</span> <span m="2643400">root</span> <span m="2643730">2</span> <span m="2643930">pi</span> <span m="2644190">n.</span> <span m="2646170">All</span> <span m="2646300">right?</span> <span m="2646430">Because</span> <span m="2647580">that</span> <span m="2647830">e to the</span> <span m="2648270">epsilon</span> <span m="2648740">n</span> <span m="2649020">goes</span> <span m="2649280">to</span> <span m="2649380">1</span> <span m="2649710">in</span> <span m="2649840">the</span> <span m="2649910">limit.</span> <span m="2650610">So</span> <span m="2650820">it</span> <span m="2650910">sort</span> <span m="2651060">of</span> <span m="2651140">disappears</span> <span m="2651770">when</span> <span m="2651870">you</span> <span m="2651930">go</span> <span m="2652060">to</span> <span m="2652130">the</span> <span m="2652220">tilde</span> <span m="2652540">notation.</span> <span m="2654490">And</span> <span m="2654700">this</span> <span m="2654910">in</span> <span m="2655010">fact</span> <span m="2655400">is</span> <span m="2655510">a</span> <span m="2655580">lower</span> <span m="2655860">bound</span> <span m="2656610">on</span> <span m="2656780">n</span> <span m="2656900">factorial,</span> <span m="2657510">and</span> <span m="2657660">very</span> <span m="2657900">close</span> <span m="2658210">to</span> <span m="2658280">the</span> <span m="2658350">right</span> <span m="2658550">answer.</span> </p>
<p><span m="2661510">Now</span> <span m="2662200">this</span> <span m="2663280">is</span> <span m="2663370">sort of an</span> <span m="2663650">amazing</span> <span m="2664530">formula.</span> <span m="2665010">I</span> <span m="2665080">mean</span> <span m="2665260">who</span> <span m="2665400">would</span> <span m="2665570">think</span> <span m="2666600">that</span> <span m="2666730">when</span> <span m="2666820">you</span> <span m="2666920">look</span> <span m="2667090">at</span> <span m="2667170">n</span> <span m="2667360">factorial</span> <span m="2669010">that</span> <span m="2669180">the</span> <span m="2669270">expression</span> <span m="2669850">would</span> <span m="2669970">have</span> <span m="2670330">e</span> <span m="2671200">and</span> <span m="2671380">pi</span> <span m="2672640">in</span> <span m="2672760">the</span> <span m="2672850">expression</span> <span m="2673300">for</span> <span m="2673560">it?</span> <span m="2674120">Maybe</span> <span m="2674540">e.</span> <span m="2674930">Maybe.</span> <span m="2675640">But</span> <span m="2675820">what's</span> <span m="2676060">pi</span> <span m="2676510">doing</span> <span m="2677560">in</span> <span m="2677690">n</span> <span m="2677820">factorial?</span> <span m="2679330">It's</span> <span m="2679580">sort</span> <span m="2679850">of</span> <span m="2679930">a</span> <span m="2680380">bizarre</span> <span m="2680730">thing.</span> </p>
<p><span m="2683410">OK.</span> <span m="2684470">That's</span> <span m="2684760">it</span> <span m="2684980">for</span> <span m="2685140">products.</span> <span m="2685660">We're</span> <span m="2685740">just</span> <span m="2685900">going</span> <span m="2686040">to do</span> <span m="2686110">the</span> <span m="2686190">one</span> <span m="2686400">example,</span> <span m="2686890">because</span> <span m="2687400">really</span> <span m="2687840">any</span> <span m="2688020">product</span> <span m="2688430">just</span> <span m="2688590">becomes</span> <span m="2688910">a</span> <span m="2688970">sum</span> <span m="2689450">through</span> <span m="2689600">the</span> <span m="2689700">log.</span> <span m="2690530">And</span> <span m="2690630">that</span> <span m="2690750">makes</span> <span m="2690960">it</span> <span m="2691050">really</span> <span m="2691300">easy.</span> <span m="2692590">Any</span> <span m="2692810">questions</span> <span m="2693540">on</span> <span m="2694100">sums</span> <span m="2694510">and</span> <span m="2694640">products?</span> <span m="2696900">Because</span> <span m="2697020">I'm</span> <span m="2697080">going</span> <span m="2697210">to</span> <span m="2697280">totally</span> <span m="2697680">change</span> <span m="2698040">gears</span> <span m="2698430">now</span> <span m="2699740">and</span> <span m="2700670">talk</span> <span m="2700940">about</span> <span m="2701580">more</span> <span m="2701790">things</span> <span m="2702080">like</span> <span m="2702270">tilde.</span> <span m="2705340">OK.</span> </p>
<p><span m="2710170">So</span> <span m="2710430">we're</span> <span m="2710530">going</span> <span m="2710650">to</span> <span m="2710720">talk</span> <span m="2711680">today</span> <span m="2712270">and</span> <span m="2712540">tomorrow</span> <span m="2712940">in</span> <span m="2713040">recitation</span> <span m="2715340">about</span> <span m="2715800">asymptotic</span> <span m="2716700">notation.</span> <span m="2718690">It is</span> <span m="2719090">used</span> <span m="2719640">all</span> <span m="2720130">over</span> <span m="2720310">the</span> <span m="2720390">place</span> <span m="2720840">in</span> <span m="2720940">computer</span> <span m="2721320">science.</span> <span m="2724130">We've</span> <span m="2724410">already</span> <span m="2724740">seen</span> <span m="2725050">one</span> <span m="2725270">example,</span> <span m="2725740">which</span> <span m="2725950">is</span> <span m="2726050">the</span> <span m="2726150">tilde</span> <span m="2726500">notation.</span> <span m="2728810">And</span> <span m="2729020">it</span> <span m="2729100">turns</span> <span m="2729460">out</span> <span m="2729760">there's</span> <span m="2729970">five</span> <span m="2730570">more</span> <span m="2730890">things</span> <span m="2731420">like</span> <span m="2731740">that</span> <span m="2733150">which</span> <span m="2733530">we're</span> <span m="2733660">going</span> <span m="2733780">to</span> <span m="2734150">learn</span> <span m="2734410">what</span> <span m="2734570">they</span> <span m="2734760">are</span> <span m="2734990">and</span> <span m="2735170">how</span> <span m="2735330">they</span> <span m="2735450">work.</span> </p>
<p><span m="2742790">They're</span> <span m="2742940">used</span> <span m="2743400">to</span> <span m="2743840">explain</span> <span m="2744650">or</span> <span m="2745490">describe</span> <span m="2746190">how</span> <span m="2746530">a</span> <span m="2746590">function</span> <span m="2747160">grows</span> <span m="2748140">in</span> <span m="2748280">the</span> <span m="2748360">limit.</span> <span m="2750330">So</span> <span m="2750560">for</span> <span m="2750680">example,</span> <span m="2751190">we</span> <span m="2751310">had</span> <span m="2751520">the</span> <span m="2751590">tilde</span> <span m="2751930">notation.</span> <span m="2754540">And</span> <span m="2754700">we</span> <span m="2754810">wrote</span> <span m="2755830">f</span> <span m="2756080">of</span> <span m="2756200">x</span> <span m="2756670">is</span> <span m="2756930">tilde</span> <span m="2757360">g</span> <span m="2757660">of</span> <span m="2757790">x</span> <span m="2761100">if</span> <span m="2761610">the</span> <span m="2761690">limit</span> <span m="2763730">as</span> <span m="2764130">x</span> <span m="2764410">goes</span> <span m="2764680">to</span> <span m="2764790">infinity</span> <span m="2766240">of</span> <span m="2766610">f</span> <span m="2768130">over</span> <span m="2768410">g</span> <span m="2770600">is</span> <span m="2770740">1.</span> </p>
<p><span m="2773880">Now</span> <span m="2774620">the</span> <span m="2774740">next</span> <span m="2775110">one--</span> <span m="2775290">and</span> <span m="2775360">most</span> <span m="2775630">commonly</span> <span m="2776280">used</span> <span m="2776610">one</span> <span m="2776850">in</span> <span m="2776940">computer</span> <span m="2777290">science--</span> <span m="2777810">is</span> <span m="2777910">called</span> <span m="2778280">the</span> <span m="2778470">O</span> <span m="2778720">notation,</span> <span m="2779910">or</span> <span m="2780130">the</span> <span m="2780230">big</span> <span m="2780560">O.</span> <span m="2783200">And</span> <span m="2783400">it's</span> <span m="2783600">written</span> <span m="2783980">like</span> <span m="2784250">this.</span> <span m="2784760">f</span> <span m="2785000">of</span> <span m="2785130">x</span> <span m="2786090">equals</span> <span m="2787150">O</span> <span m="2787880">of</span> <span m="2788410">g</span> <span m="2788650">of</span> <span m="2788780">x.</span> <span m="2791100">And</span> <span m="2791290">it</span> <span m="2791440">means</span> <span m="2794330">that</span> <span m="2794610">the</span> <span m="2794700">limit</span> <span m="2796310">as</span> <span m="2796570">x</span> <span m="2796840">goes</span> <span m="2797100">to</span> <span m="2797230">infinity</span> <span m="2798890">of</span> <span m="2799120">the</span> <span m="2799290">absolute</span> <span m="2799740">value</span> <span m="2800210">of</span> <span m="2800370">f</span> <span m="2800560">of</span> <span m="2800710">x</span> <span m="2801840">over</span> <span m="2802110">g</span> <span m="2802320">of</span> <span m="2802430">x</span> <span m="2805020">is</span> <span m="2805930">convergence--</span> <span m="2806510">is</span> <span m="2806700">less than</span> <span m="2807110">infinity.</span> <span m="2808210">So</span> <span m="2808380">it's</span> <span m="2808500">finite,</span> <span m="2812156">and</span> <span m="2812560">it</span> <span m="2812710">can't</span> <span m="2812990">diverge.</span> </p>
<p><span m="2815760">The</span> <span m="2815930">interpretation</span> <span m="2816840">is</span> <span m="2817130">that</span> <span m="2818050">this</span> <span m="2818400">function</span> <span m="2820730">is</span> <span m="2821330">up</span> <span m="2821520">to</span> <span m="2821590">constant</span> <span m="2822130">factors</span> <span m="2823120">upper-bounded</span> <span m="2823740">by</span> <span m="2823880">this</span> <span m="2824100">one,</span> <span m="2825860">that</span> <span m="2826590">this</span> <span m="2826970">grows</span> <span m="2828450">the</span> <span m="2828570">same</span> <span m="2828970">rate</span> <span m="2829300">or</span> <span m="2829430">slower</span> <span m="2830420">than</span> <span m="2830590">this</span> <span m="2830780">one</span> <span m="2830970">grows</span> <span m="2831690">as</span> <span m="2831900">x</span> <span m="2832080">gets</span> <span m="2832340">large.</span> <span m="2833880">Because</span> <span m="2834240">if</span> <span m="2834730">f</span> <span m="2834960">was</span> <span m="2835110">growing</span> <span m="2835360">a</span> <span m="2835420">lot</span> <span m="2835640">faster</span> <span m="2836080">than</span> <span m="2836240">g</span> <span m="2837040">in</span> <span m="2837160">the</span> <span m="2837230">limit,</span> <span m="2838450">then</span> <span m="2838640">the</span> <span m="2838860">limit</span> <span m="2839140">would</span> <span m="2839250">be</span> <span m="2839340">the</span> <span m="2839460">ratio</span> <span m="2839800">is</span> <span m="2839890">infinity.</span> <span m="2840520">And</span> <span m="2840630">we're</span> <span m="2840720">saying</span> <span m="2841040">that</span> <span m="2841260">doesn't</span> <span m="2841520">happen.</span> <span m="2842560">All right.</span> <span m="2843020">I'm</span> <span m="2843100">not</span> <span m="2843200">going</span> <span m="2843320">to do</span> <span m="2843400">a</span> <span m="2843470">lot</span> <span m="2843800">of</span> <span m="2843870">examples</span> <span m="2845040">for</span> <span m="2845200">this.</span> <span m="2849600">In</span> <span m="2849820">fact,</span> <span m="2850150">let's</span> <span m="2850200">go</span> <span m="2850360">over</span> <span m="2850510">here.</span> </p>
<p><span m="2851400">There's</span> <span m="2851580">several</span> <span m="2852060">other</span> <span m="2852280">ways</span> <span m="2852620">you'll</span> <span m="2852730">see</span> <span m="2852930">it</span> <span m="2853060">written.</span> <span m="2870980">For</span> <span m="2871130">example,</span> <span m="2872380">you'll</span> <span m="2872530">see</span> <span m="2872730">people</span> <span m="2873040">write</span> <span m="2873460">f</span> <span m="2873690">of</span> <span m="2873830">x</span> <span m="2874610">is</span> <span m="2874880">less</span> <span m="2875300">than</span> <span m="2875560">or</span> <span m="2875670">equal</span> <span m="2875960">to</span> <span m="2876920">O</span> <span m="2877270">of</span> <span m="2877350">g</span> <span m="2877570">of</span> <span m="2877690">x,</span> <span m="2879390">because</span> <span m="2880180">the</span> <span m="2880280">natural</span> <span m="2880620">interpretation</span> <span m="2881290">is</span> <span m="2881550">f</span> <span m="2881790">is</span> <span m="2881930">not</span> <span m="2882210">growing</span> <span m="2882490">faster</span> <span m="2882940">than</span> <span m="2883100">g--</span> <span m="2883460">sort</span> <span m="2883660">of</span> <span m="2883780">upper-bounded</span> <span m="2884390">by</span> <span m="2884540">g.</span> <span m="2885950">You'll</span> <span m="2886160">see</span> <span m="2886360">people</span> <span m="2886730">write</span> <span m="2887760">f</span> <span m="2887980">of</span> <span m="2888120">x</span> <span m="2888580">is</span> <span m="2889570">O</span> <span m="2889840">of</span> <span m="2889880">g</span> <span m="2890130">of</span> <span m="2890250">x.</span> </p>
<p><span m="2893060">And</span> <span m="2893390">the</span> <span m="2893450">formal</span> <span m="2894040">math</span> <span m="2894460">way</span> <span m="2894690">that</span> <span m="2895380">people</span> <span m="2895660">don't</span> <span m="2895980">use,</span> <span m="2896440">but</span> <span m="2896610">it's</span> <span m="2896770">the</span> <span m="2896840">formally</span> <span m="2897310">correct</span> <span m="2897700">one,</span> <span m="2898590">is</span> <span m="2898980">f</span> <span m="2899150">of</span> <span m="2899290">x</span> <span m="2899650">is</span> <span m="2899790">an</span> <span m="2899900">element</span> <span m="2900760">of</span> <span m="2901110">O</span> <span m="2901380">of</span> <span m="2901650">g</span> <span m="2901870">of</span> <span m="2901980">x.</span> <span m="2903750">And</span> <span m="2903890">the</span> <span m="2903960">interpretation</span> <span m="2904760">there</span> <span m="2905110">is</span> <span m="2905330">that</span> <span m="2905970">O</span> <span m="2906250">of</span> <span m="2906330">g</span> <span m="2906560">of</span> <span m="2906680">x</span> <span m="2906890">is</span> <span m="2906980">a</span> <span m="2907040">set</span> <span m="2907350">of</span> <span m="2907460">functions</span> <span m="2907960">that</span> <span m="2908090">don't</span> <span m="2908370">grow</span> <span m="2908570">any</span> <span m="2908700">faster</span> <span m="2909100">than</span> <span m="2909250">g.</span> <span m="2909950">And</span> <span m="2910210">f</span> <span m="2910430">is</span> <span m="2910540">one</span> <span m="2910750">of</span> <span m="2910830">them.</span> <span m="2911810">You</span> <span m="2911970">can</span> <span m="2912150">use</span> <span m="2912580">any</span> <span m="2912820">of</span> <span m="2912910">these</span> <span m="2913650">and</span> <span m="2913810">that's</span> <span m="2914000">fine.</span> <span m="2914710">People</span> <span m="2915090">will</span> <span m="2915250">know</span> <span m="2915400">what you</span> <span m="2915490">mean,</span> <span m="2915740">and</span> <span m="2915980">it'll be</span> <span m="2916150">fine</span> <span m="2916400">for</span> <span m="2916520">the</span> <span m="2916590">class.</span> </p>
<p><span m="2918860">All right.</span> <span m="2919060">Let's</span> <span m="2919240">see</span> <span m="2919460">some</span> <span m="2919650">examples,</span> <span m="2922240">and</span> <span m="2922370">see</span> <span m="2922520">how</span> <span m="2922640">to</span> <span m="2922750">prove</span> <span m="2923650">a</span> <span m="2923760">big</span> <span m="2923970">O</span> <span m="2924140">relation.</span> <span m="2927300">Let</span> <span m="2927990">f</span> <span m="2928260">of</span> <span m="2928400">x</span> <span m="2929850">be</span> <span m="2929980">the</span> <span m="2930050">function</span> <span m="2930460">x,</span> <span m="2931000">and</span> <span m="2931270">g</span> <span m="2931480">of</span> <span m="2931600">x</span> <span m="2932670">be</span> <span m="2932830">the</span> <span m="2932920">function</span> <span m="2933490">x</span> <span m="2933740">squared.</span> <span m="2935670">Then</span> <span m="2937500">in</span> <span m="2937610">this</span> <span m="2937780">case</span> <span m="2938170">f</span> <span m="2938410">of</span> <span m="2938540">x</span> <span m="2938920">is</span> <span m="2939340">O</span> <span m="2939590">of</span> <span m="2939690">g</span> <span m="2939900">of</span> <span m="2940020">x--</span> <span m="2942910">because</span> <span m="2943840">x</span> <span m="2944130">doesn't</span> <span m="2944410">grow</span> <span m="2944520">any</span> <span m="2944640">faster</span> <span m="2945020">than</span> <span m="2945140">x</span> <span m="2945280">squared</span> <span m="2945690">as</span> <span m="2945820">x</span> <span m="2946010">gets</span> <span m="2946250">large.</span> </p>
<p><span m="2946940">To</span> <span m="2947100">prove</span> <span m="2947430">it,</span> <span m="2949810">we</span> <span m="2949990">take</span> <span m="2950260">the</span> <span m="2950340">limit</span> <span m="2951030">as</span> <span m="2951260">x</span> <span m="2951510">goes</span> <span m="2951780">to</span> <span m="2951890">infinity</span> <span m="2954115">of</span> <span m="2954370">x</span> <span m="2955570">over</span> <span m="2956120">x</span> <span m="2956330">squared.</span> <span m="2958700">That</span> <span m="2959160">is</span> <span m="2959650">equal</span> <span m="2960000">to</span> <span m="2960360">what?</span> <span m="2961040">What is</span> <span m="2961140">the</span> <span m="2961220">limit</span> <span m="2961610">of</span> <span m="2961770">x</span> <span m="2961980">over</span> <span m="2962170">x</span> <span m="2962290">squared</span> <span m="2962550">as</span> <span m="2962660">x</span> <span m="2962810">goes</span> <span m="2962980">to</span> <span m="2963060">infinity?</span> <span m="2965920">0.</span> <span m="2968350">And</span> <span m="2968580">that</span> <span m="2968810">is</span> <span m="2969010">less</span> <span m="2969260">than</span> <span m="2969360">infinity.</span> <span m="2972000">So</span> <span m="2972230">we</span> <span m="2972350">proved</span> <span m="2972650">it.</span> </p>
<p><span m="2973960">So</span> <span m="2974160">if</span> <span m="2974240">you</span> <span m="2974330">get</span> <span m="2974500">asked</span> <span m="2975060">to</span> <span m="2975160">prove</span> <span m="2976850">f</span> <span m="2977500">is</span> <span m="2977930">O</span> <span m="2978360">of</span> <span m="2978790">g,</span> <span m="2979650">what</span> <span m="2979890">you</span> <span m="2979970">do</span> <span m="2980350">is</span> <span m="2980430">you</span> <span m="2980510">take</span> <span m="2980770">the</span> <span m="2980850">limit</span> <span m="2981585">of</span> <span m="2981840">f</span> <span m="2982210">over</span> <span m="2982460">g</span> <span m="2983810">as</span> <span m="2983990">x</span> <span m="2984140">goes</span> <span m="2984320">to</span> <span m="2984400">infinity</span> <span m="2984860">and</span> <span m="2984990">show</span> <span m="2985360">that</span> <span m="2985620">it</span> <span m="2985730">doesn't</span> <span m="2986060">diverge.</span> </p>
<p><span m="2987871">All</span> <span m="2988280">right.</span> <span m="2988530">Let's</span> <span m="2988710">do</span> <span m="2990190">another</span> <span m="2990450">example.</span> <span m="2992258">In</span> <span m="2992710">this</span> <span m="2992910">case</span> <span m="2993170">we'll</span> <span m="2993270">show</span> <span m="2993590">that</span> <span m="2994140">f</span> <span m="2994370">is</span> <span m="2994500">not</span> <span m="2994840">O</span> <span m="2994980">of</span> <span m="2995080">g.</span> <span m="3002270">x</span> <span m="3002530">squared</span> <span m="3003120">is</span> <span m="3003380">not</span> <span m="3004350">O</span> <span m="3004710">of x.</span> <span m="3006232">Of</span> <span m="3006640">course,</span> <span m="3006890">x</span> <span m="3007080">squared</span> <span m="3007380">grows</span> <span m="3007610">faster</span> <span m="3008020">than</span> <span m="3008160">x,</span> <span m="3008360">but</span> <span m="3008470">let's</span> <span m="3008670">see</span> <span m="3009130">why</span> <span m="3009380">not.</span> </p>
<p><span m="3012670">Well</span> <span m="3012790">we</span> <span m="3012880">take</span> <span m="3013160">the</span> <span m="3013240">limit</span> <span m="3015100">as</span> <span m="3015270">x</span> <span m="3015470">goes</span> <span m="3015650">to</span> <span m="3015710">infinity</span> <span m="3016010">of</span> <span m="3017210">x</span> <span m="3017470">squared</span> <span m="3018820">over</span> <span m="3019170">x.</span> <span m="3021130">And</span> <span m="3021280">what</span> <span m="3021430">is</span> <span m="3021560">that</span> <span m="3021800">limit?</span> <span m="3023920">Infinity.</span> <span m="3025870">And</span> <span m="3026030">so</span> <span m="3026500">it</span> <span m="3026670">can't</span> <span m="3027000">be</span> <span m="3027100">true</span> <span m="3027360">that</span> <span m="3027480">it's</span> <span m="3028100">x squared</span> <span m="3028360">is O</span> <span m="3028620">of</span> <span m="3028850">x,</span> <span m="3029120">because</span> <span m="3029310">it</span> <span m="3029380">has</span> <span m="3029530">to</span> <span m="3029590">be</span> <span m="3029690">less</span> <span m="3029980">than</span> <span m="3030080">infinity.</span> <span m="3031541">Yeah?</span> </p>
<p><span m="3032496">AUDIENCE: Sorry.</span> <span m="3032962">I missed it.</span> <span m="3033894">What's alpha?</span> </p>
<p><span m="3037720">PROFESSOR: Infinity.</span> <span m="3041180">Sorry.</span> <span m="3041770">Infinity.</span> <span m="3042050">There we</span> <span m="3042110">go.</span> <span m="3042350">That's</span> <span m="3042550">important</span> <span m="3042990">in</span> <span m="3043060">the</span> <span m="3043120">definition.</span> <span m="3043710">Yep.</span> <span m="3044770">Good.</span> <span m="3049150">All right.</span> <span m="3049390">What</span> <span m="3049540">about</span> <span m="3049790">this?</span> <span m="3052190">Is</span> <span m="3053180">x</span> <span m="3053450">squared</span> <span m="3054260">equal</span> <span m="3055580">O</span> <span m="3056110">of</span> <span m="3056800">a</span> <span m="3056900">million</span> <span m="3057630">times</span> <span m="3058995">x?</span> <span m="3064750">Yes?</span> <span m="3065630">No?</span> <span m="3067920">What</span> <span m="3068050">do</span> <span m="3068110">you</span> <span m="3068190">think?</span> <span m="3072100">This</span> <span m="3072360">looks</span> <span m="3072670">pretty</span> <span m="3072910">darn</span> <span m="3073200">big.</span> </p>
<p><span m="3074166">AUDIENCE: It's a</span> <span m="3074650">constant.</span> </p>
<p><span m="3075470">PROFESSOR: That</span> <span m="3075740">a</span> <span m="3075790">constant.</span> <span m="3077750">So</span> <span m="3078170">in</span> <span m="3078250">the</span> <span m="3078330">limit,</span> <span m="3078910">as</span> <span m="3079130">x</span> <span m="3079350">goes</span> <span m="3079610">to</span> <span m="3079730">infinity,</span> <span m="3081090">this</span> <span m="3082310">over</span> <span m="3082600">that</span> <span m="3083670">is</span> <span m="3083890">infinite.</span> <span m="3085710">All right.</span> <span m="3086040">So</span> <span m="3087150">no.</span> <span m="3088720">Because</span> <span m="3089190">the</span> <span m="3089280">limit</span> <span m="3090722">of</span> <span m="3091220">x</span> <span m="3091510">goes</span> <span m="3091740">to</span> <span m="3091830">infinity</span> <span m="3092250">of</span> <span m="3092370">x</span> <span m="3092550">squared</span> <span m="3093580">over</span> <span m="3094020">10</span> <span m="3094290">to</span> <span m="3094370">the</span> <span m="3094490">sixth</span> <span m="3094870">x,</span> <span m="3096850">that</span> <span m="3097090">equals</span> <span m="3097390">infinity.</span> <span m="3100180">So</span> <span m="3101630">the</span> <span m="3101740">thing</span> <span m="3101940">with</span> <span m="3102070">this</span> <span m="3102320">big O</span> <span m="3102730">notation</span> <span m="3103300">is</span> <span m="3103380">you</span> <span m="3103450">just</span> <span m="3103680">ignore</span> <span m="3104080">our</span> <span m="3104310">constant</span> <span m="3104540">factors.</span> <span m="3105390">They</span> <span m="3105520">don't</span> <span m="3105810">do</span> <span m="3106040">anything.</span> <span m="3106650">Just</span> <span m="3106870">forget</span> <span m="3107190">about</span> <span m="3107510">them.</span> </p>
<p><span m="3111690">What</span> <span m="3111820">about</span> <span m="3112150">this?</span> <span m="3112690">Is</span> <span m="3115520">10</span> <span m="3115920">to</span> <span m="3116000">the</span> <span m="3116100">sixth</span> <span m="3116670">x</span> <span m="3117230">equal</span> <span m="3118240">O</span> <span m="3118660">of</span> <span m="3118900">x</span> <span m="3119364">squared?</span> <span m="3123170">Yes.</span> <span m="3125010">Because</span> <span m="3125620">you</span> <span m="3125710">just</span> <span m="3125950">ignore</span> <span m="3126300">the</span> <span m="3126390">constant.</span> <span m="3126930">That's</span> <span m="3127110">what</span> <span m="3127220">the</span> <span m="3127300">big</span> <span m="3127520">O</span> <span m="3127670">is</span> <span m="3127790">for,</span> <span m="3128090">and</span> <span m="3128180">why</span> <span m="3128360">it's</span> <span m="3128490">used</span> <span m="3128720">all</span> <span m="3128820">the</span> <span m="3128890">time</span> <span m="3129140">in</span> <span m="3129200">computer</span> <span m="3129520">science.</span> <span m="3130810">So</span> <span m="3131220">yes,</span> <span m="3132510">that's</span> <span m="3132730">true.</span> </p>
<p><span m="3133670">The</span> <span m="3134170">proof</span> <span m="3134550">is</span> <span m="3134690">very</span> <span m="3134970">simple.</span> <span m="3136260">The</span> <span m="3136340">limit</span> <span m="3136960">as</span> <span m="3137190">x</span> <span m="3137460">goes</span> <span m="3137810">to</span> <span m="3137990">infinity</span> <span m="3139410">of</span> <span m="3139540">100</span> <span m="3141220">x</span> <span m="3141470">squared--</span> <span m="3142260">whoops.</span> <span m="3143280">I</span> <span m="3143430">could even</span> <span m="3143710">put</span> <span m="3143840">a</span> <span m="3143870">square</span> <span m="3144180">there.</span> <span m="3145110">10</span> <span m="3145300">to</span> <span m="3145370">the</span> <span m="3145430">sixth</span> <span m="3145730">x</span> <span m="3145940">squared</span> <span m="3147710">over</span> <span m="3147990">x</span> <span m="3148190">squared</span> <span m="3149260">is</span> <span m="3149730">10</span> <span m="3150040">to</span> <span m="3150130">the</span> <span m="3150200">sixth.</span> <span m="3151780">That's</span> <span m="3152100">less</span> <span m="3152590">than infinity.</span> </p>
<p><span m="3157000">In</span> <span m="3157110">fact,</span> <span m="3157390">I</span> <span m="3157460">could</span> <span m="3157620">take</span> <span m="3157880">any</span> <span m="3158180">quadratic</span> <span m="3158740">polynomial.</span> <span m="3160380">x</span> <span m="3160590">squared</span> <span m="3161240">plus</span> <span m="3162290">100x</span> <span m="3163770">plus</span> <span m="3164200">10</span> <span m="3164460">to</span> <span m="3164550">the</span> <span m="3164610">seventh</span> <span m="3166460">is O of</span> <span m="3166780">x</span> <span m="3166940">squared.</span> <span m="3169970">Because</span> <span m="3170160">smaller</span> <span m="3170590">order</span> <span m="3170840">terms</span> <span m="3171260">don't</span> <span m="3171470">matter.</span> <span m="3172140">The</span> <span m="3172270">limit</span> <span m="3172660">of</span> <span m="3172930">this</span> <span m="3174030">here</span> <span m="3174480">over</span> <span m="3174720">that</span> <span m="3175890">in</span> <span m="3175980">that</span> <span m="3176100">case</span> <span m="3176390">is</span> <span m="3176490">1.</span> <span m="3176880">So</span> <span m="3177040">it's</span> <span m="3177210">less</span> <span m="3177420">than</span> <span m="3177520">infinity</span> <span m="3178356">as x</span> <span m="3178774">gets big.</span> </p>
<p><span m="3182050">All</span> <span m="3182310">right</span> <span m="3182590">let's</span> <span m="3182800">do</span> <span m="3183550">a</span> <span m="3183580">few</span> <span m="3183780">more.</span> <span m="3189200">This</span> <span m="3189460">really</span> <span m="3189710">is</span> <span m="3189800">not</span> <span m="3190000">so</span> <span m="3190200">hard.</span> <span m="3190660">It's</span> <span m="3190810">a</span> <span m="3190860">little</span> <span m="3191170">dull.</span> <span m="3192770">But</span> <span m="3194190">you've</span> <span m="3194330">got</span> <span m="3194440">to</span> <span m="3194510">know</span> <span m="3194800">it</span> <span m="3195820">for</span> <span m="3195940">later.</span> <span m="3201780">OK.</span> <span m="3201960">I</span> <span m="3202030">could</span> <span m="3202180">take</span> <span m="3202370">classes</span> <span m="3202730">like</span> <span m="3202890">6046,</span> <span m="3203860">and</span> <span m="3203970">you'll</span> <span m="3204430">never</span> <span m="3204740">see</span> <span m="3204970">any</span> <span m="3205120">constants,</span> <span m="3205610">because</span> <span m="3205850">everything</span> <span m="3206350">is</span> <span m="3207200">things</span> <span m="3207440">like</span> <span m="3207650">big</span> <span m="3207760">O</span> <span m="3208040">and</span> <span m="3208370">the</span> <span m="3208440">other</span> <span m="3208580">notation</span> <span m="3209070">we'll</span> <span m="3209160">talk</span> <span m="3209420">about.</span> </p>
<p><span m="3211040">What</span> <span m="3211170">about</span> <span m="3211440">this?</span> <span m="3212050">X</span> <span m="3212420">to</span> <span m="3212490">the</span> <span m="3212570">10th</span> <span m="3215140">equal</span> <span m="3215980">O</span> <span m="3216350">of</span> <span m="3216720">e</span> <span m="3216970">to the</span> <span m="3217110">x.</span> <span m="3220720">Is</span> <span m="3221070">x</span> <span m="3221410">to the</span> <span m="3221500">10</span> <span m="3221970">O</span> <span m="3222240">of</span> <span m="3222340">e to</span> <span m="3222590">the</span> <span m="3222750">x?</span> <span m="3226390">Yeah,</span> <span m="3226940">it</span> <span m="3227120">is.</span> <span m="3228740">So</span> <span m="3229110">this</span> <span m="3229260">is</span> <span m="3229330">a</span> <span m="3229370">theorem.</span> <span m="3233250">Proof.</span> <span m="3235240">Limit</span> <span m="3235570">as</span> <span m="3235750">x</span> <span m="3236030">goes</span> <span m="3236190">to</span> <span m="3236340">infinity.</span> <span m="3237730">x</span> <span m="3238060">to</span> <span m="3238140">the</span> <span m="3238210">10</span> <span m="3238830">over</span> <span m="3239030">e</span> <span m="3239270">to</span> <span m="3239330">the</span> <span m="3239470">x.</span> <span m="3240680">That</span> <span m="3241030">equals</span> <span m="3241380">0,</span> <span m="3241920">which</span> <span m="3242130">is</span> <span m="3242220">certainly</span> <span m="3242570">less than</span> <span m="3242870">infinity.</span> </p>
<p><span m="3246090">All right.</span> <span m="3246310">Now</span> <span m="3246430">the</span> <span m="3246560">reason</span> <span m="3246920">this</span> <span m="3247040">stuff</span> <span m="3247270">is</span> <span m="3247400">useful</span> <span m="3248070">when</span> <span m="3248240">you're</span> <span m="3248330">doing</span> <span m="3249180">the</span> <span m="3249330">analysis</span> <span m="3249770">of</span> <span m="3249890">algorithms</span> <span m="3251310">is</span> <span m="3251520">because</span> <span m="3251820">when</span> <span m="3251930">you're</span> <span m="3252510">talking</span> <span m="3252890">about</span> <span m="3253140">things</span> <span m="3253590">like</span> <span m="3254000">matrix</span> <span m="3254450">multiplication</span> <span m="3255140">algorithms</span> <span m="3255690">or</span> <span m="3255800">things</span> <span m="3256130">like</span> <span m="3256390">that,</span> <span m="3257410">the</span> <span m="3257560">actual</span> <span m="3258020">running</span> <span m="3258470">time</span> <span m="3259380">depends</span> <span m="3259790">on</span> <span m="3259900">the</span> <span m="3259960">machine</span> <span m="3260460">you're</span> <span m="3260580">dealing</span> <span m="3260940">with--</span> <span m="3262350">the</span> <span m="3262450">machine</span> <span m="3262810">characteristics--</span> <span m="3264450">details</span> <span m="3264950">in</span> <span m="3265060">how</span> <span m="3265210">you</span> <span m="3265330">count</span> <span m="3265630">steps</span> <span m="3266130">of</span> <span m="3266220">the</span> <span m="3266360">algorithm.</span> <span m="3267320">For</span> <span m="3267430">example,</span> <span m="3267810">doing</span> <span m="3268140">matrix</span> <span m="3268480">multiplication,</span> <span m="3268805">do</span> <span m="3269130">you</span> <span m="3269250">count</span> <span m="3269560">multiply</span> <span m="3270130">different</span> <span m="3270480">than</span> <span m="3270620">you</span> <span m="3270710">count</span> <span m="3271110">plus</span> <span m="3271820">addition?</span> </p>
<p><span m="3273160">And</span> <span m="3273910">if</span> <span m="3274090">you</span> <span m="3274170">start</span> <span m="3274500">getting</span> <span m="3274910">caught</span> <span m="3275270">up</span> <span m="3275410">in</span> <span m="3275490">all</span> <span m="3275810">those</span> <span m="3276040">details</span> <span m="3277440">you</span> <span m="3277620">lose</span> <span m="3277880">the</span> <span m="3277990">important</span> <span m="3278460">fact</span> <span m="3278830">of the</span> <span m="3278960">algorithm.</span> <span m="3279880">For</span> <span m="3279980">example,</span> <span m="3280340">matrix</span> <span m="3280700">multiplication--</span> <span m="3281440">the</span> <span m="3281620">elementary</span> <span m="3282150">algorithm</span> <span m="3282510">takes</span> <span m="3283230">order</span> <span m="3284170">O</span> <span m="3284470">of</span> <span m="3285140">n</span> <span m="3285400">cubed</span> <span m="3285660">steps.</span> <span m="3287320">And</span> <span m="3287780">that's</span> <span m="3288010">the</span> <span m="3288110">important</span> <span m="3288510">fact.</span> <span m="3288930">And it</span> <span m="3289260">tells</span> <span m="3289620">you</span> <span m="3289740">that</span> <span m="3289870">if</span> <span m="3289980">you</span> <span m="3290800">double</span> <span m="3291120">the</span> <span m="3291220">size</span> <span m="3291590">of</span> <span m="3291650">the</span> <span m="3291710">matrix</span> <span m="3293070">the</span> <span m="3293210">running</span> <span m="3293520">time</span> <span m="3293810">will</span> <span m="3293920">increase</span> <span m="3294320">by</span> <span m="3294470">at</span> <span m="3294560">most</span> <span m="3294920">a</span> <span m="3294960">factor</span> <span m="3295320">of</span> <span m="3295400">eight</span> <span m="3295960">in</span> <span m="3296140">the</span> <span m="3296220">limit</span> <span m="3297210">as</span> <span m="3297460">n</span> <span m="3297600">gets</span> <span m="3297830">large.</span> </p>
<p><span m="3298890">And</span> <span m="3298940">so</span> <span m="3299050">you</span> <span m="3299170">might</span> <span m="3299680">write</span> <span m="3300020">things</span> <span m="3300340">like</span> <span m="3301720">the</span> <span m="3301820">time</span> <span m="3303300">to</span> <span m="3303470">multiply</span> <span m="3307630">n</span> <span m="3307860">by</span> <span m="3308020">n</span> <span m="3308150">matrices.</span> <span m="3312810">You</span> <span m="3312920">might</span> <span m="3313100">call</span> <span m="3313360">that</span> <span m="3313830">t of</span> <span m="3313860">n,</span> <span m="3314910">and</span> <span m="3315030">then</span> <span m="3315170">state</span> <span m="3315485">that</span> <span m="3315800">is</span> <span m="3316030">O</span> <span m="3316290">of</span> <span m="3316780">n</span> <span m="3316990">cubed.</span> <span m="3319530">All</span> <span m="3319690">right.</span> <span m="3319890">And</span> <span m="3320070">you</span> <span m="3320140">don't</span> <span m="3320280">have</span> <span m="3320390">to</span> <span m="3320460">worry</span> <span m="3320630">about</span> <span m="3320830">what</span> <span m="3321520">computer</span> <span m="3321970">it's</span> <span m="3322110">running</span> <span m="3322410">on,</span> <span m="3323785">exactly</span> <span m="3324200">how</span> <span m="3324340">you're</span> <span m="3324470">counting</span> <span m="3324820">the</span> <span m="3324880">details</span> <span m="3325380">of</span> <span m="3325460">what</span> <span m="3325590">a</span> <span m="3325630">step</span> <span m="3326010">is</span> <span m="3326250">or what</span> <span m="3326500">time</span> <span m="3326840">is.</span> <span m="3327580">The</span> <span m="3327690">important</span> <span m="3328200">thing</span> <span m="3328880">is</span> <span m="3329240">that</span> <span m="3329340">it's</span> <span m="3329480">growing</span> <span m="3330350">at</span> <span m="3330540">most</span> <span m="3331130">as</span> <span m="3331280">a</span> <span m="3331340">cubic</span> <span m="3332310">of</span> <span m="3332470">the</span> <span m="3332550">size</span> <span m="3333650">of</span> <span m="3333900">the</span> <span m="3333970">matrices.</span> </p>
<p><span m="3335490">And</span> <span m="3335680">so</span> <span m="3335960">in</span> <span m="3336200">6046</span> <span m="3337300">and</span> <span m="3337820">6006</span> <span m="3339730">you</span> <span m="3339940">will</span> <span m="3340150">spend</span> <span m="3340450">lots</span> <span m="3340720">of</span> <span m="3340810">time</span> <span m="3341730">proving</span> <span m="3342190">that</span> <span m="3342360">things</span> <span m="3342680">are</span> <span m="3342975">the</span> <span m="3343270">running</span> <span m="3343670">time,</span> <span m="3344040">or</span> <span m="3344130">the</span> <span m="3344220">size</span> <span m="3345290">is</span> <span m="3345590">O</span> <span m="3345980">of some</span> <span m="3346820">function</span> <span m="3347230">of</span> <span m="3347320">the</span> <span m="3347650">input</span> <span m="3348010">size.</span> <span m="3350345">All right.</span> </p>
<p><span m="3353780">Any</span> <span m="3353960">questions?</span> <span m="3355340">Yeah.</span> </p>
<p><span m="3356260">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3365692">PROFESSOR: Yeah.</span> <span m="3366150">It gets</span> <span m="3366270">all</span> <span m="3366600">dicey</span> <span m="3367220">if</span> <span m="3367310">you</span> <span m="3367410">do</span> <span m="3367710">oscillatory</span> <span m="3368260">things.</span> <span m="3368580">And it</span> <span m="3368650">depends</span> <span m="3369180">how</span> <span m="3369340">you</span> <span m="3369460">define</span> <span m="3369850">it.</span> <span m="3370350">So</span> <span m="3370520">we're</span> <span m="3370660">going</span> <span m="3370780">to</span> <span m="3370870">stay</span> <span m="3371170">away</span> <span m="3371430">from</span> <span m="3371620">that</span> <span m="3373360">and</span> <span m="3373610">just</span> <span m="3374450">stick</span> <span m="3374800">with</span> <span m="3374920">this</span> <span m="3375170">definition.</span> <span m="3375730">So</span> <span m="3375950">if</span> <span m="3376040">this</span> <span m="3376200">guy</span> <span m="3376440">oscillates--</span> <span m="3377780">like</span> <span m="3378030">f</span> <span m="3378280">is</span> <span m="3378390">a</span> <span m="3378930">sine</span> <span m="3379170">of</span> <span m="3379280">x,</span> <span m="3380130">sine</span> <span m="3380480">of</span> <span m="3380580">x</span> <span m="3380790">is</span> <span m="3380930">O</span> <span m="3381060">of</span> <span m="3381170">x.</span> <span m="3381865">All right.</span> <span m="3382160">It</span> <span m="3382400">doesn't</span> <span m="3382640">have</span> <span m="3382950">to</span> <span m="3383190">converge.</span> <span m="3384060">But</span> <span m="3384280">as</span> <span m="3384420">long</span> <span m="3384700">as</span> <span m="3384930">the</span> <span m="3385050">ratio</span> <span m="3385710">is</span> <span m="3385850">less</span> <span m="3386230">than</span> <span m="3386500">infinity.</span> </p>
<p><span m="3387380">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3389580">PROFESSOR: Yeah.</span> <span m="3390110">Everything</span> <span m="3390520">you</span> <span m="3390790">do.</span> <span m="3391740">Maybe</span> <span m="3391980">you'll</span> <span m="3392090">see</span> <span m="3392230">sine</span> <span m="3392520">functions</span> <span m="3392870">sometimes.</span> <span m="3393250">But</span> <span m="3393360">everything</span> <span m="3393590">you</span> <span m="3393650">do</span> <span m="3393780">in</span> <span m="3393860">algorithms</span> <span m="3394320">is</span> <span m="3394440">going</span> <span m="3394570">to</span> <span m="3394640">be</span> <span m="3394790">growing</span> <span m="3395320">monotonically.</span> <span m="3396640">Yeah.</span> <span m="3396850">Nothing--</span> <span m="3397610">no</span> <span m="3397870">wacky</span> <span m="3398430">stuff</span> <span m="3398740">in</span> <span m="3398830">math</span> <span m="3399120">usually.</span> <span m="3400660">Any</span> <span m="3400870">other</span> <span m="3401020">questions?</span> <span m="3402790">Yeah.</span> </p>
<p><span m="3403555">AUDIENCE: [INAUDIBLE]</span> <span m="3405010">sort of</span> <span m="3405495">comparing</span> <span m="3405980">something that's</span> <span m="3406465">the same</span> <span m="3407435">magnitude</span> <span m="3407920">that you go</span> <span m="3408405">off</span> <span m="3408890">into infinity.</span> </p>
<p><span m="3409870">PROFESSOR: Yeah.</span> <span m="3410450">It's</span> <span m="3410740">upper</span> <span m="3410980">bounded.</span> <span m="3412110">So</span> <span m="3412230">it's</span> <span m="3412320">not</span> <span m="3412460">the</span> <span m="3412530">same.</span> <span m="3412880">We'll</span> <span m="3412960">get</span> <span m="3413080">to</span> <span m="3413160">the</span> <span m="3413250">same in a</span> <span m="3413570">minute.</span> <span m="3414040">But</span> <span m="3414140">it's</span> <span m="3414210">upper</span> <span m="3414500">bounded.</span> <span m="3415310">f</span> <span m="3415600">is</span> <span m="3415750">upper</span> <span m="3415960">bounded</span> <span m="3416320">by</span> <span m="3416470">g</span> <span m="3417120">up</span> <span m="3417340">to</span> <span m="3417880">a</span> <span m="3417970">fixed</span> <span m="3418250">constant</span> <span m="3418950">as</span> <span m="3419140">you</span> <span m="3419210">go</span> <span m="3419310">to</span> <span m="3419410">infinity.</span> <span m="3420820">And</span> <span m="3420980">all</span> <span m="3421210">constants</span> <span m="3421740">get</span> <span m="3421910">washed</span> <span m="3422390">away</span> <span m="3422800">by</span> <span m="3422940">O.</span> <span m="3424450">Yeah.</span> </p>
<p><span m="3425366">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3428850">PROFESSOR: Ah,</span> <span m="3429210">why</span> <span m="3429480">is</span> <span m="3429620">this</span> <span m="3429800">true?</span> <span m="3430840">Well</span> <span m="3432280">technically</span> <span m="3432590">we'd</span> <span m="3432800">have</span> <span m="3432920">to</span> <span m="3432980">go</span> <span m="3433080">back</span> <span m="3433270">to</span> <span m="3433350">calculus</span> <span m="3433920">and</span> <span m="3434010">use</span> <span m="3434170">L'Hopital's</span> <span m="3434790">rule</span> <span m="3435090">or</span> <span m="3435160">something</span> <span m="3435430">like</span> <span m="3435630">that,</span> <span m="3436510">because</span> <span m="3437280">this</span> <span m="3437510">function</span> <span m="3437930">exponential</span> <span m="3438570">grows</span> <span m="3438910">a</span> <span m="3438980">lot</span> <span m="3439400">faster</span> <span m="3440520">than</span> <span m="3440660">x</span> <span m="3440820">to</span> <span m="3440880">the</span> <span m="3440940">tenth.</span> </p>
<p><span m="3441370">So</span> <span m="3441655">L'Hopital's</span> <span m="3441940">rule</span> <span m="3442170">is</span> <span m="3442270">you</span> <span m="3442360">take</span> <span m="3442560">the</span> <span m="3442630">derivative</span> <span m="3443010">of</span> <span m="3443100">this</span> <span m="3443380">over</span> <span m="3443490">the</span> <span m="3443560">derivative of</span> <span m="3443920">that.</span> <span m="3444900">Then</span> <span m="3445030">you'd</span> <span m="3445150">have</span> <span m="3445370">x</span> <span m="3445570">to</span> <span m="3445670">the</span> <span m="3445770">ninth</span> <span m="3446630">over</span> <span m="3446900">e</span> <span m="3447010">to</span> <span m="3447070">the</span> <span m="3447180">x.</span> <span m="3447850">You</span> <span m="3448010">do</span> <span m="3448180">it</span> <span m="3448270">nine</span> <span m="3448540">more</span> <span m="3448730">times.</span> <span m="3449780">And</span> <span m="3450070">you'd</span> <span m="3450210">get</span> <span m="3450660">10</span> <span m="3450920">factorial</span> <span m="3451670">over</span> <span m="3451890">e</span> <span m="3452060">to the</span> <span m="3452190">x.</span> <span m="3453150">And</span> <span m="3453740">that</span> <span m="3453970">limit</span> <span m="3454170">is</span> <span m="3454270">zero.</span> <span m="3455000">So</span> <span m="3455170">that's</span> <span m="3455390">how</span> <span m="3455510">you'd</span> <span m="3455680">actually</span> <span m="3456000">do</span> <span m="3456270">it</span> <span m="3456480">using</span> <span m="3456860">L'Hopital's</span> <span m="3457300">rule.</span> <span m="3459150">Yep.</span> <span m="3460240">Any</span> <span m="3460500">polynomial</span> <span m="3461120">grows</span> <span m="3461320">slower</span> <span m="3461700">than</span> <span m="3461900">any</span> <span m="3462130">exponential.</span> </p>
<p><span m="3465380">All</span> <span m="3465550">right.</span> <span m="3466090">Let's</span> <span m="3467260">now</span> <span m="3467430">do</span> <span m="3467560">some</span> <span m="3467970">examples</span> <span m="3468590">that</span> <span m="3468910">can</span> <span m="3469410">get</span> <span m="3469560">you</span> <span m="3470020">screwed</span> <span m="3470480">up.</span> <span m="3473800">What</span> <span m="3473990">about</span> <span m="3474630">this</span> <span m="3474930">one?</span> <span m="3476790">Is</span> <span m="3477110">4</span> <span m="3477600">to the</span> <span m="3477800">x</span> <span m="3479905">O</span> <span m="3480390">to</span> <span m="3480710">2 to the</span> <span m="3480860">x?</span> <span m="3486900">Is</span> <span m="3487030">that</span> <span m="3487220">true?</span> <span m="3490180">No,</span> <span m="3490610">it's</span> <span m="3490790">not</span> <span m="3491100">true.</span> </p>
<p><span m="3491830">I</span> <span m="3491910">mean,</span> <span m="3492390">the</span> <span m="3492490">mistake</span> <span m="3492980">you</span> <span m="3493100">might</span> <span m="3493440">make</span> <span m="3493710">is</span> <span m="3493800">say,</span> <span m="3494090">oh,</span> <span m="3494320">4</span> <span m="3494680">and</span> <span m="3495050">2,</span> <span m="3495420">4</span> <span m="3495690">is</span> <span m="3495790">only</span> <span m="3495970">double</span> <span m="3496290">2.</span> <span m="3496940">Big</span> <span m="3497300">O wipes</span> <span m="3497630">out</span> <span m="3497780">constant</span> <span m="3498180">factors.</span> <span m="3498750">So</span> <span m="3498920">the</span> <span m="3499050">answer</span> <span m="3499280">is</span> <span m="3499340">yes.</span> <span m="3500360">That</span> <span m="3500580">would</span> <span m="3500680">be</span> <span m="3500780">bad.</span> </p>
<p><span m="3501740">Better</span> <span m="3502170">to</span> <span m="3502700">look</span> <span m="3502910">at</span> <span m="3502990">the</span> <span m="3503070">limit.</span> <span m="3504770">So</span> <span m="3504790">we'll</span> <span m="3504910">state</span> <span m="3505150">the</span> <span m="3505220">correct</span> <span m="3505560">theorem--</span> <span m="3506810">4</span> <span m="3507140">to the</span> <span m="3507310">x</span> <span m="3507760">is</span> <span m="3508040">not</span> <span m="3509140">O</span> <span m="3509400">of</span> <span m="3509460">2</span> <span m="3509780">to the</span> <span m="3509940">x.</span> <span m="3510970">And</span> <span m="3511100">the</span> <span m="3511170">proof</span> <span m="3511670">is</span> <span m="3512450">obtained</span> <span m="3513040">by</span> <span m="3513330">looking</span> <span m="3513580">at</span> <span m="3513660">the</span> <span m="3513740">limit</span> <span m="3515590">as</span> <span m="3515800">x</span> <span m="3516030">goes</span> <span m="3516240">to</span> <span m="3516330">infinity</span> <span m="3516850">of</span> <span m="3517140">4</span> <span m="3517390">to</span> <span m="3517450">the</span> <span m="3517680">x</span> <span m="3518000">over</span> <span m="3518340">2</span> <span m="3518660">to the</span> <span m="3518800">x.</span> <span m="3520060">That</span> <span m="3520370">is</span> <span m="3520580">just</span> <span m="3520860">the</span> <span m="3520960">limit</span> <span m="3522400">4</span> <span m="3522640">to the</span> <span m="3522730">x</span> <span m="3522890">over</span> <span m="3523020">2</span> <span m="3523290">to the</span> <span m="3523400">x</span> <span m="3523660">is</span> <span m="3523880">2 to</span> <span m="3524140">the</span> <span m="3524270">x.</span> <span m="3525000">That's</span> <span m="3525350">infinity.</span> <span m="3527730">So</span> <span m="3527880">the</span> <span m="3527960">limit</span> <span m="3528230">is</span> <span m="3528370">not</span> <span m="3529080">bounded.</span> <span m="3529556">All right?</span> <span m="3530510">So</span> <span m="3530890">it</span> <span m="3530960">is</span> <span m="3531080">not</span> <span m="3531850">big</span> <span m="3532000">O</span> <span m="3532210">there.</span> </p>
<p><span m="3533701">All</span> <span m="3534200">right.</span> <span m="3534380">Here</span> <span m="3534570">is</span> <span m="3534670">one.</span> <span m="3537770">Is</span> <span m="3538410">10</span> <span m="3539560">O</span> <span m="3539830">of</span> <span m="3539900">1?</span> <span m="3545230">I've</span> <span m="3545580">purposely</span> <span m="3546170">written</span> <span m="3546430">it</span> <span m="3546530">this</span> <span m="3546720">way.</span> <span m="3548790">Yes,</span> <span m="3549860">it</span> <span m="3550060">is.</span> <span m="3550940">And</span> <span m="3551200">in</span> <span m="3551290">fact</span> <span m="3551710">the</span> <span m="3551820">interpretation</span> <span m="3552760">here</span> <span m="3553770">is</span> <span m="3554000">that</span> <span m="3554100">you've</span> <span m="3554210">got</span> <span m="3554500">some</span> <span m="3554790">function</span> <span m="3555320">f</span> <span m="3555570">of</span> <span m="3555710">x</span> <span m="3556020">that's</span> <span m="3556160">just</span> <span m="3556410">always</span> <span m="3556830">equal</span> <span m="3557060">to</span> <span m="3557180">10,</span> <span m="3558310">and</span> <span m="3558490">some</span> <span m="3558790">function</span> <span m="3559230">g</span> <span m="3559500">of</span> <span m="3559610">x</span> <span m="3560640">that's</span> <span m="3560840">always</span> <span m="3561300">equal</span> <span m="3561540">to</span> <span m="3561620">1.</span> </p>
<p><span m="3562550">And</span> <span m="3562750">then</span> <span m="3562920">it</span> <span m="3563010">is</span> <span m="3563180">true</span> <span m="3563490">in</span> <span m="3563570">this</span> <span m="3563740">case</span> <span m="3564090">that</span> <span m="3564270">f</span> <span m="3564450">of</span> <span m="3564590">x</span> <span m="3565340">is</span> <span m="3565560">O  of g</span> <span m="3565940">of</span> <span m="3566000">x.</span> <span m="3569010">All</span> <span m="3569250">right. But</span> <span m="3569450">any</span> <span m="3569730">constant</span> <span m="3570520">function</span> <span m="3571730">is</span> <span m="3571990">O</span> <span m="3572415">of</span> <span m="3572840">any</span> <span m="3573100">other</span> <span m="3573260">constant</span> <span m="3573810">function.</span> <span m="3576020">We're</span> <span m="3576190">going</span> <span m="3576320">to</span> <span m="3576380">revisit</span> <span m="3576810">this</span> <span m="3577010">in</span> <span m="3577090">a</span> <span m="3577130">nastier</span> <span m="3577690">scenario</span> <span m="3578190">in</span> <span m="3578250">a</span> <span m="3578300">little</span> <span m="3578490">bit.</span> </p>
<p><span m="3581100">OK.</span> <span m="3585490">Sometimes</span> <span m="3586640">you'll</span> <span m="3586830">see</span> <span m="3588210">notation</span> <span m="3588810">like</span> <span m="3589020">this.</span> <span m="3590670">H</span> <span m="3590970">of</span> <span m="3591090">n--</span> <span m="3591660">in</span> <span m="3592060">fact,</span> <span m="3592400">we</span> <span m="3592500">saw</span> <span m="3592540">this</span> <span m="3592750">earlier--</span> <span m="3593890">is</span> <span m="3594360">the</span> <span m="3594480">log</span> <span m="3594870">of</span> <span m="3594990">n</span> <span m="3595966">plus</span> <span m="3596432">delta</span> <span m="3597364">plus</span> <span m="3597830">O of</span> <span m="3598860">1</span> <span m="3599080">over</span> <span m="3599380">n.</span> <span m="3600770">When</span> <span m="3600930">we</span> <span m="3601050">wrote</span> <span m="3601270">out</span> <span m="3601460">the</span> <span m="3601890">formula</span> <span m="3602550">for</span> <span m="3602680">the</span> <span m="3602770">n'th</span> <span m="3603000">harmonic</span> <span m="3603400">number,</span> <span m="3604310">it</span> <span m="3604480">was</span> <span m="3604740">log of</span> <span m="3604980">n</span> <span m="3605460">plus</span> <span m="3605740">Euler's</span> <span m="3606130">constant</span> <span m="3607100">plus</span> <span m="3608260">something</span> <span m="3608650">like</span> <span m="3608850">1</span> <span m="3609160">over</span> <span m="3609450">n,</span> <span m="3609790">constant</span> <span m="3610230">times</span> <span m="3610430">1</span> <span m="3610600">over</span> <span m="3610850">n.</span> <span m="3611200">And</span> <span m="3611370">then</span> <span m="3611530">some</span> <span m="3611790">even</span> <span m="3612070">smaller</span> <span m="3612680">terms.</span> </p>
<p><span m="3613750">And</span> <span m="3613910">so</span> <span m="3614030">people</span> <span m="3614370">will</span> <span m="3614530">write</span> <span m="3614750">it</span> <span m="3614860">like</span> <span m="3615070">this.</span> <span m="3615390">And</span> <span m="3615480">it</span> <span m="3615600">tells</span> <span m="3616050">you</span> <span m="3616250">that</span> <span m="3616870">the</span> <span m="3617060">error</span> <span m="3617410">terms</span> <span m="3618960">grow</span> <span m="3619180">no</span> <span m="3619350">faster</span> <span m="3619840">than</span> <span m="3619960">a</span> <span m="3620000">constant</span> <span m="3620440">times</span> <span m="3620640">1</span> <span m="3620790">over</span> <span m="3621030">n.</span> <span m="3622260">Technically</span> <span m="3623560">this</span> <span m="3623790">is</span> <span m="3623890">the</span> <span m="3623990">wrong</span> <span m="3624310">way</span> <span m="3624440">to</span> <span m="3624550">write</span> <span m="3624810">it.</span> <span m="3625400">But</span> <span m="3625630">everybody</span> <span m="3626150">does</span> <span m="3626430">it,</span> <span m="3626620">so</span> <span m="3626950">you</span> <span m="3627080">can</span> <span m="3627400">too.</span> </p>
<p><span m="3629260">What</span> <span m="3629450">it</span> <span m="3629560">means</span> <span m="3630370">is</span> <span m="3630620">this.</span> <span m="3631320">H</span> <span m="3631600">of</span> <span m="3631730">n</span> <span m="3631910">minus</span> <span m="3632310">ln</span> <span m="3632470">of</span> <span m="3632590">n</span> <span m="3632850">minus</span> <span m="3633190">delta</span> <span m="3634300">is</span> <span m="3634630">O</span> <span m="3635190">of</span> <span m="3635450">1</span> <span m="3635680">over</span> <span m="3635970">n.</span> <span m="3636980">All right.</span> <span m="3637380">That's</span> <span m="3637940">the</span> <span m="3638030">correct</span> <span m="3638470">way</span> <span m="3638600">to</span> <span m="3638710">write</span> <span m="3638990">it,</span> <span m="3640050">because</span> <span m="3640350">you're</span> <span m="3640440">taking</span> <span m="3640770">the</span> <span m="3640880">ratio</span> <span m="3641510">of</span> <span m="3641610">this</span> <span m="3642100">over</span> <span m="3642320">that.</span> <span m="3643790">This</span> <span m="3644240">is</span> <span m="3644360">how</span> <span m="3644530">people</span> <span m="3644820">will</span> <span m="3644960">do</span> <span m="3645180">it.</span> <span m="3645950">All right.</span> <span m="3646040">So</span> <span m="3646080">just</span> <span m="3646855">make</span> <span m="3647220">sure</span> <span m="3647320">you</span> <span m="3647390">understand</span> <span m="3647940">that,</span> <span m="3649670">what</span> <span m="3649900">it</span> <span m="3649990">means</span> <span m="3650260">when</span> <span m="3650400">you</span> <span m="3650460">put</span> <span m="3650640">it</span> <span m="3650720">over</span> <span m="3650890">there.</span> </p>
<p><span m="3652070">Same</span> <span m="3652340">thing</span> <span m="3652470">with tilde</span> <span m="3652570">notation.</span> <span m="3654150">People</span> <span m="3654470">will</span> <span m="3654610">write</span> <span m="3654930">this--</span> <span m="3655820">H</span> <span m="3656100">of</span> <span m="3656210">n</span> <span m="3656340">is</span> <span m="3656490">tilde</span> <span m="3657780">ln of</span> <span m="3658200">n</span> <span m="3659410">plus</span> <span m="3659860">Euler's</span> <span m="3660230">constant.</span> <span m="3661360">The</span> <span m="3661490">right</span> <span m="3661870">way</span> <span m="3662000">to</span> <span m="3662110">write</span> <span m="3662350">it</span> <span m="3663640">is</span> <span m="3663840">Hn</span> <span m="3664140">minus</span> <span m="3664570">Euler's</span> <span m="3664860">constant</span> <span m="3665360">is</span> <span m="3665480">tilde</span> <span m="3666390">ln</span> <span m="3666600">of</span> <span m="3666680">n.</span> </p>
<p><span m="3668720">Now</span> <span m="3668810">can</span> <span m="3668960">anybody</span> <span m="3669310">tell</span> <span m="3669610">me</span> <span m="3669750">why</span> <span m="3671490">this</span> <span m="3672320">is</span> <span m="3672560">really</span> <span m="3672950">not</span> <span m="3673190">a</span> <span m="3673240">good</span> <span m="3673430">thing</span> <span m="3673600">to</span> <span m="3673670">do?</span> <span m="3674570">And</span> <span m="3674660">you</span> <span m="3674720">think</span> <span m="3675010">of</span> <span m="3675150">how</span> <span m="3675520">things</span> <span m="3675890">get</span> <span m="3676060">really</span> <span m="3676320">screwed</span> <span m="3676660">up</span> <span m="3677440">if</span> <span m="3677610">you</span> <span m="3677750">allow</span> <span m="3677990">to</span> <span m="3678070">start</span> <span m="3678350">doing</span> <span m="3678620">this?</span> <span m="3681300">Can</span> <span m="3681510">you</span> <span m="3681770">think of</span> <span m="3681860">anything</span> <span m="3682220">that--?</span> </p>
<p><span m="3685620">What if</span> <span m="3685730">I</span> <span m="3685760">did</span> <span m="3685960">this?</span> <span m="3686280">Is this</span> <span m="3686600">true?</span> <span m="3691530">10.</span> <span m="3695070">The</span> <span m="3695360">n'th</span> <span m="3695530">harmonic</span> <span m="3696060">is</span> <span m="3696700">harmonic</span> <span m="3697090">number</span> <span m="3697400">is</span> <span m="3697690">tilde</span> <span m="3697830">of log</span> <span m="3698180">of</span> <span m="3698290">n</span> <span m="3698500">plus</span> <span m="3698830">10.</span> <span m="3701180">Is</span> <span m="3701300">that</span> <span m="3701440">true?</span> <span m="3704280">Think</span> <span m="3704560">about</span> <span m="3704810">that.</span> </p>
<p><span m="3706050">People are</span> <span m="3706320">shaking</span> <span m="3706650">their</span> <span m="3706790">heads.</span> <span m="3708260">I</span> <span m="3708290">mean</span> <span m="3708500">how</span> <span m="3708750">can</span> <span m="3709000">it</span> <span m="3709070">be</span> <span m="3709150">true?</span> <span m="3709330">Because</span> <span m="3709640">we</span> <span m="3709750">know</span> <span m="3710010">this</span> <span m="3710240">is.</span> <span m="3711090">We know that's</span> <span m="3711580">what</span> <span m="3711690">it</span> <span m="3711780">is.</span> </p>
<p><span m="3713700">In</span> <span m="3713800">fact,</span> <span m="3714140">this</span> <span m="3714200">is</span> <span m="3714300">true.</span> <span m="3715730">I</span> <span m="3715850">can</span> <span m="3715990">even</span> <span m="3716160">make</span> <span m="3716360">it</span> <span m="3717455">a</span> <span m="3717860">million.</span> <span m="3719000">10</span> <span m="3719180">to</span> <span m="3719240">the</span> <span m="3719320">sixth</span> <span m="3719580">here.</span> <span m="3720470">It's</span> <span m="3720650">true.</span> <span m="3722530">Because</span> <span m="3724110">the</span> <span m="3724250">ratio</span> <span m="3724760">of</span> <span m="3725070">this</span> <span m="3725490">over</span> <span m="3725780">that</span> <span m="3726800">in</span> <span m="3726920">the</span> <span m="3726990">limit</span> <span m="3728020">is</span> <span m="3728190">what?</span> <span m="3731160">1.</span> <span m="3732360">And</span> <span m="3732490">that's</span> <span m="3732700">all</span> <span m="3732840">I</span> <span m="3732910">need</span> <span m="3733090">for</span> <span m="3733190">tilde.</span> <span m="3736140">All</span> <span m="3736300">right.</span> <span m="3736590">So</span> <span m="3737470">when</span> <span m="3737670">I</span> <span m="3737800">wrote</span> <span m="3738110">this,</span> <span m="3738690">everybody</span> <span m="3739280">knew</span> <span m="3739490">what</span> <span m="3739650">I</span> <span m="3739750">meant.</span> </p>
<p><span m="3741730">But</span> <span m="3741930">I</span> <span m="3742020">could</span> <span m="3742180">write</span> <span m="3742400">this</span> <span m="3742600">and</span> <span m="3742700">be</span> <span m="3742770">just</span> <span m="3743030">as</span> <span m="3743110">true</span> <span m="3743280">mathematically.</span> <span m="3744960">And</span> <span m="3745070">so</span> <span m="3745210">you've</span> <span m="3745290">got</span> <span m="3745460">to</span> <span m="3745520">be</span> <span m="3745630">sort</span> <span m="3745840">of</span> <span m="3745910">careful</span> <span m="3747150">about</span> <span m="3747430">this.</span> <span m="3747640">Really,</span> <span m="3747900">the</span> <span m="3748120">right</span> <span m="3748430">way</span> <span m="3748530">to</span> <span m="3748620">do</span> <span m="3748770">it</span> <span m="3748850">is</span> <span m="3749000">this,</span> <span m="3749840">because</span> <span m="3750170">now</span> <span m="3750380">we're</span> <span m="3750490">taking</span> <span m="3750860">off</span> <span m="3751090">that</span> <span m="3751260">term,</span> <span m="3751880">and</span> <span m="3752060">then it's--</span> <span m="3752570">actually</span> <span m="3752800">I</span> <span m="3752830">did</span> <span m="3752980">this</span> <span m="3753210">wrong.</span> <span m="3753820">I</span> <span m="3753910">should've</span> <span m="3754100">put</span> <span m="3754230">the</span> <span m="3754360">[INAUDIBLE].</span> <span m="3754860">Whoops.</span> <span m="3755200">I</span> <span m="3755260">should've</span> <span m="3755470">done</span> <span m="3755620">it this</span> <span m="3755820">way.</span> <span m="3756210">Let</span> <span m="3756320">me</span> <span m="3756410">rewrite</span> <span m="3756730">that.</span> </p>
<p><span m="3757490">The</span> <span m="3757610">right</span> <span m="3757810">way</span> <span m="3757910">to</span> <span m="3757980">do</span> <span m="3758090">it</span> <span m="3758150">is</span> <span m="3758240">H</span> <span m="3758340">of n--</span> <span m="3759140">yeah--</span> <span m="3760080">is</span> <span m="3760370">this.</span> <span m="3760710">You</span> <span m="3760790">put</span> <span m="3760960">the</span> <span m="3761010">small</span> <span m="3761460">term</span> <span m="3761790">out</span> <span m="3761930">over</span> <span m="3762100">here.</span> <span m="3762560">That's</span> <span m="3762840">the</span> <span m="3762920">right</span> <span m="3763130">way.</span> <span m="3763740">Because</span> <span m="3763940">now</span> <span m="3764200">it's</span> <span m="3764340">not</span> <span m="3764680">true</span> <span m="3765160">that</span> <span m="3765290">it's</span> <span m="3765420">10</span> <span m="3765630">to</span> <span m="3765710">the</span> <span m="3765790">sixth</span> <span m="3766120">out</span> <span m="3766380">here--</span> <span m="3767670">all</span> <span m="3767860">right--</span> <span m="3768210">for</span> <span m="3768350">the</span> <span m="3768450">ratio</span> <span m="3768830">of</span> <span m="3768920">that</span> <span m="3769230">over</span> <span m="3770100">this</span> <span m="3770350">guy.</span> </p>
<p><span m="3772180">All</span> <span m="3772340">right.</span> <span m="3772550">So</span> <span m="3772910">you</span> <span m="3773070">can</span> <span m="3773250">do</span> <span m="3773500">it.</span> <span m="3773740">But</span> <span m="3773920">be</span> <span m="3774060">careful,</span> <span m="3775010">because</span> <span m="3775370">you</span> <span m="3775470">might</span> <span m="3775750">write</span> <span m="3776060">wrong</span> <span m="3776420">things</span> <span m="3776810">that</span> <span m="3777290">technically</span> <span m="3777790">would</span> <span m="3777940">fit</span> <span m="3778160">the</span> <span m="3778230">definition.</span> <span m="3779400">Same</span> <span m="3779690">thing</span> <span m="3780010">with</span> <span m="3780860">big</span> <span m="3781070">O.</span> <span m="3781380">Same</span> <span m="3782090">problem</span> <span m="3782400">there.</span> <span m="3783040">Any</span> <span m="3783240">questions</span> <span m="3785419">about</span> <span m="3785892">that?</span> </p>
<p><span m="3790160">All</span> <span m="3790370">right.</span> <span m="3792260">All</span> <span m="3792430">right.</span> <span m="3792580">Now</span> <span m="3792730">one</span> <span m="3793040">thing</span> <span m="3793810">that</span> <span m="3793970">people</span> <span m="3794250">do</span> <span m="3794360">all</span> <span m="3794530">the</span> <span m="3794620">time</span> <span m="3794950">that</span> <span m="3795070">you</span> <span m="3795170">cannot</span> <span m="3795720">do</span> <span m="3797460">is</span> <span m="3797780">to</span> <span m="3797910">use</span> <span m="3798370">big</span> <span m="3798690">O</span> <span m="3799000">as</span> <span m="3799110">a</span> <span m="3799190">lower</span> <span m="3799661">bound.</span> <span m="3802760">All</span> <span m="3803130">right.</span> <span m="3803370">So</span> <span m="3803470">that's</span> <span m="3803710">one</span> <span m="3803970">abuse</span> <span m="3804390">that</span> <span m="3804660">we</span> <span m="3804840">won't--</span> <span m="3805495">you'll</span> <span m="3805860">get</span> <span m="3806170">marked</span> <span m="3806470">wrong</span> <span m="3806710">for</span> <span m="3806830">doing</span> <span m="3807090">that,</span> <span m="3809050">even</span> <span m="3809350">though</span> <span m="3810260">famous</span> <span m="3810710">researchers</span> <span m="3811200">will</span> <span m="3811310">do</span> <span m="3811490">it,</span> <span m="3811590">and you'll</span> <span m="3811710">probably</span> <span m="3812150">even</span> <span m="3812370">have</span> <span m="3812520">a</span> <span m="3812580">professor</span> <span m="3813010">do</span> <span m="3813170">it</span> <span m="3813270">sometime.</span> </p>
<p><span m="3817280">So</span> <span m="3817530">you</span> <span m="3817720">may</span> <span m="3817910">not</span> <span m="3818280">do</span> <span m="3818770">this.</span> <span m="3819760">f of</span> <span m="3820090">x</span> <span m="3820610">is</span> <span m="3820840">bigger</span> <span m="3821210">than</span> <span m="3821410">or</span> <span m="3821530">equal</span> <span m="3821810">to</span> <span m="3822210">O of</span> <span m="3822410">g</span> <span m="3822680">of</span> <span m="3822790">x.</span> <span m="3824270">That</span> <span m="3824500">is</span> <span m="3824660">meaningless.</span> <span m="3830180">Because</span> <span m="3830520">the</span> <span m="3830610">whole</span> <span m="3830850">point</span> <span m="3831200">of</span> <span m="3831280">big</span> <span m="3831705">O</span> <span m="3832480">is</span> <span m="3832690">that</span> <span m="3832790">it's</span> <span m="3832920">an</span> <span m="3833080">upper</span> <span m="3833170">bound.</span> <span m="3834440">So</span> <span m="3834620">you</span> <span m="3834730">can't</span> <span m="3835430">use</span> <span m="3835680">it</span> <span m="3835750">as</span> <span m="3835840">a</span> <span m="3835900">lower</span> <span m="3836120">bound.</span> </p>
<p><span m="3837230">And</span> <span m="3837560">the</span> <span m="3837650">reason</span> <span m="3837920">you</span> <span m="3837990">can't</span> <span m="3838310">do</span> <span m="3838500">it</span> <span m="3838580">is</span> <span m="3838670">because</span> <span m="3839200">there</span> <span m="3839420">is</span> <span m="3840880">another</span> <span m="3841230">symbol</span> <span m="3842030">to</span> <span m="3842160">do</span> <span m="3842310">what</span> <span m="3842480">you</span> <span m="3842570">want</span> <span m="3842760">to</span> <span m="3842820">do.</span> <span m="3843820">And</span> <span m="3844030">that's</span> <span m="3844240">called</span> <span m="3844550">the</span> <span m="3844660">omega</span> <span m="3845030">symbol.</span> <span m="3847960">And</span> <span m="3848440">it's</span> <span m="3848600">written</span> <span m="3848970">like</span> <span m="3849210">this.</span> <span m="3852380">It's</span> <span m="3852560">a</span> <span m="3852610">capital</span> <span m="3853050">omega.</span> <span m="3855810">And</span> <span m="3856190">it's if</span> <span m="3856510">the</span> <span m="3856610">limit</span> <span m="3857930">as</span> <span m="3858090">x</span> <span m="3858340">goes</span> <span m="3858620">to</span> <span m="3858740">infinity</span> <span m="3860534">of</span> <span m="3861021">f</span> <span m="3861510">of</span> <span m="3861650">x</span> <span m="3862020">over</span> <span m="3863720">g</span> <span m="3864020">of</span> <span m="3864130">x</span> <span m="3865080">is</span> <span m="3865410">greater</span> <span m="3865800">than</span> <span m="3865970">zero.</span> </p>
<p><span m="3868640">OK.</span> <span m="3869210">So</span> <span m="3869420">now</span> <span m="3869880">we're</span> <span m="3869990">getting</span> <span m="3870240">the</span> <span m="3870310">lower</span> <span m="3870730">bound</span> <span m="3871060">version.</span> <span m="3871500">In</span> <span m="3871580">fact</span> <span m="3871970">it's</span> <span m="3872210">just</span> <span m="3872640">the</span> <span m="3872790">opposite</span> <span m="3873906">of</span> <span m="3875590">big</span> <span m="3875790">O.</span> <span m="3877355">In</span> <span m="3877770">fact,</span> <span m="3878020">we'll</span> <span m="3878140">state</span> <span m="3878390">a</span> <span m="3878420">theorem.</span> <span m="3879080">Not</span> <span m="3879320">hard</span> <span m="3879590">to</span> <span m="3879670">prove.</span> <span m="3880060">We</span> <span m="3880160">won't</span> <span m="3880340">do</span> <span m="3880510">it.</span> <span m="3881800">f</span> <span m="3882060">of</span> <span m="3882200">x</span> <span m="3882540">equals</span> <span m="3883020">O</span> <span m="3883420">of</span> <span m="3883640">g</span> <span m="3883890">of</span> <span m="3884020">x</span> <span m="3886630">if</span> <span m="3886850">and</span> <span m="3886940">only</span> <span m="3887260">if</span> <span m="3888550">g</span> <span m="3888830">of</span> <span m="3888950">x</span> <span m="3890210">equals</span> <span m="3891400">big</span> <span m="3891660">omega</span> <span m="3893060">of</span> <span m="3893460">f</span> <span m="3893660">of</span> <span m="3893770">x.</span> </p>
<p><span m="3897370">In</span> <span m="3897470">other</span> <span m="3897630">words,</span> <span m="3898370">f</span> <span m="3898660">grows</span> <span m="3898890">no</span> <span m="3899090">faster</span> <span m="3899580">than</span> <span m="3899740">g</span> <span m="3901280">if</span> <span m="3901530">and</span> <span m="3901630">only</span> <span m="3901990">if</span> <span m="3902570">g</span> <span m="3902830">grows</span> <span m="3903440">at</span> <span m="3903590">least</span> <span m="3903900">as</span> <span m="3904030">fast</span> <span m="3904610">as</span> <span m="3904830">f.</span> <span m="3906590">One's</span> <span m="3906890">an</span> <span m="3906980">upper</span> <span m="3907160">bound</span> <span m="3907490">one.</span> <span m="3907640">One's</span> <span m="3907700">a</span> <span m="3907760">lower</span> <span m="3908040">bound.</span> <span m="3909010">And</span> <span m="3909240">you've</span> <span m="3909380">just got</span> <span m="3909520">to</span> <span m="3909590">use</span> <span m="3909830">the</span> <span m="3909900">symbol</span> <span m="3910230">the</span> <span m="3910310">right</span> <span m="3910550">way.</span> </p>
<p><span m="3911430">Now</span> <span m="3911580">you</span> <span m="3911730">can</span> <span m="3912220">do</span> <span m="3912760">this.</span> <span m="3913070">You</span> <span m="3913180">can</span> <span m="3913600">write</span> <span m="3915050">f</span> <span m="3915280">of</span> <span m="3915430">x</span> <span m="3916260">bigger</span> <span m="3916720">than</span> <span m="3916930">or</span> <span m="3917040">equal</span> <span m="3917300">to</span> <span m="3918530">omega</span> <span m="3918890">g</span> <span m="3919060">of</span> <span m="3919180">x.</span> <span m="3919620">That's</span> <span m="3919940">OK.</span> <span m="3921771">All right?</span> <span m="3922250">That's</span> <span m="3922540">all</span> <span m="3922620">right</span> <span m="3922750">to</span> <span m="3922860">do.</span> </p>
<p><span m="3924420">All right.</span> <span m="3924630">Let's</span> <span m="3924790">do</span> <span m="3924900">some</span> <span m="3925080">examples.</span> <span m="3929910">So</span> <span m="3930190">x</span> <span m="3930360">squared</span> <span m="3931020">is</span> <span m="3931570">omega</span> <span m="3932050">of</span> <span m="3932200">x,</span> <span m="3932950">because</span> <span m="3933130">it</span> <span m="3933210">grows</span> <span m="3934180">faster</span> <span m="3934590">than</span> <span m="3934720">x.</span> <span m="3936360">2</span> <span m="3936590">to</span> <span m="3936690">the</span> <span m="3936840">x</span> <span m="3938340">is</span> <span m="3938770">omega</span> <span m="3939115">of</span> <span m="3939460">x</span> <span m="3939650">squared.</span> </p>
<p><span m="3954200">All</span> <span m="3954350">right.</span> <span m="3955960">x</span> <span m="3956310">over</span> <span m="3956610">100</span> <span m="3958800">is</span> <span m="3959390">omega</span> <span m="3960150">of</span> <span m="3960280">100x</span> <span m="3962440">plus</span> <span m="3963250">25,</span> <span m="3966260">because</span> <span m="3966470">constants</span> <span m="3967170">don't</span> <span m="3967460">matter.</span> <span m="3968580">x</span> <span m="3968900">over</span> <span m="3969150">100</span> <span m="3969620">grows</span> <span m="3969850">at</span> <span m="3969970">least</span> <span m="3970330">as</span> <span m="3970480">fast</span> <span m="3970970">as</span> <span m="3971070">100x</span> <span m="3971690">does</span> <span m="3972610">in</span> <span m="3972690">terms,</span> <span m="3972920">because</span> <span m="3973080">we're</span> <span m="3973170">measuring</span> <span m="3973560">the</span> <span m="3973620">growth.</span> <span m="3974940">So</span> <span m="3975010">constants</span> <span m="3975570">go</span> <span m="3975820">away.</span> </p>
<p><span m="3977670">If</span> <span m="3977890">you</span> <span m="3978010">want</span> <span m="3978270">to</span> <span m="3978360">say</span> <span m="3978630">the</span> <span m="3978720">running</span> <span m="3979110">time</span> <span m="3979690">of</span> <span m="3980110">that</span> <span m="3980320">algorithm</span> <span m="3980830">is</span> <span m="3980960">at</span> <span m="3981070">least</span> <span m="3981570">quadratic,</span> <span m="3983230">you</span> <span m="3983500">would</span> <span m="3983700">write</span> <span m="3984240">T</span> <span m="3984510">of</span> <span m="3984630">n</span> <span m="3985590">is</span> <span m="3985970">omega</span> <span m="3986500">n</span> <span m="3986860">squared.</span> <span m="3988502">All</span> <span m="3988940">right.</span> <span m="3989190">And</span> <span m="3989280">that</span> <span m="3989350">says</span> <span m="3989600">that</span> <span m="3990220">the</span> <span m="3990350">running</span> <span m="3990570">time</span> <span m="3990770">is</span> <span m="3990860">at</span> <span m="3990990">least</span> <span m="3991420">n</span> <span m="3991610">squared.</span> <span m="3993310">Any</span> <span m="3993500">questions</span> <span m="3994030">on</span> <span m="3994650">big</span> <span m="3994810">omega?</span> <span m="3997570">Not</span> <span m="3997850">too</span> <span m="3998130">hard</span> <span m="3998370">once</span> <span m="3998550">you</span> <span m="3998640">have</span> <span m="3998780">big</span> <span m="3998990">O.</span> <span m="3999110">It's</span> <span m="3999240">just</span> <span m="3999610">the</span> <span m="3999770">opposite.</span> </p>
<p><span m="4002380">All</span> <span m="4002640">right</span> <span m="4003280">the</span> <span m="4003390">next</span> <span m="4003770">symbol</span> <span m="4004950">is</span> <span m="4006370">the</span> <span m="4006540">version</span> <span m="4006890">of</span> <span m="4006970">equality,</span> <span m="4007530">where</span> <span m="4007680">you're</span> <span m="4007790">bigger--</span> <span m="4008410">at</span> <span m="4008470">least</span> <span m="4008690">as</span> <span m="4008820">big</span> <span m="4009020">as,</span> <span m="4009570">and</span> <span m="4009750">at</span> <span m="4009830">most</span> <span m="4010140">as</span> <span m="4010260">big</span> <span m="4010490">as.</span> <span m="4011240">There's</span> <span m="4011410">a</span> <span m="4011470">special</span> <span m="4011890">symbol</span> <span m="4012550">for</span> <span m="4012690">that.</span> <span m="4015180">So</span> <span m="4015390">in</span> <span m="4015470">that</span> <span m="4015650">case</span> <span m="4015910">we</span> <span m="4016030">use</span> <span m="4016330">theta.</span> </p>
<p><span m="4022160">And</span> <span m="4022290">we</span> <span m="4022400">say</span> <span m="4022690">that</span> <span m="4022970">f</span> <span m="4023230">of</span> <span m="4023370">x</span> <span m="4024250">equals</span> <span m="4024840">theta</span> <span m="4026210">of</span> <span m="4026300">g of</span> <span m="4026720">x</span> <span m="4028750">if</span> <span m="4029230">the</span> <span m="4029320">limit</span> <span m="4031550">is</span> <span m="4032040">both</span> <span m="4033160">bigger</span> <span m="4033460">than</span> <span m="4033630">0</span> <span m="4035130">and</span> <span m="4035900">less</span> <span m="4036270">than</span> <span m="4036430">infinity.</span> <span m="4039360">All</span> <span m="4039510">right.</span> <span m="4039780">So</span> <span m="4039950">it's</span> <span m="4040050">just</span> <span m="4040210">a</span> <span m="4040280">shorthand</span> <span m="4040830">way</span> <span m="4041030">of</span> <span m="4041150">saying</span> <span m="4042250">big</span> <span m="4042550">O</span> <span m="4043050">and</span> <span m="4043620">big</span> <span m="4043810">omega.</span> <span m="4044550">That's</span> <span m="4044730">what</span> <span m="4044860">theta</span> <span m="4045110">means.</span> <span m="4045400">It</span> <span m="4045490">just</span> <span m="4045740">both</span> <span m="4046000">hold</span> <span m="4046270">true.</span> <span m="4048970">All right.</span> <span m="4049220">And that's</span> <span m="4049450">a</span> <span m="4049490">theorem</span> <span m="4049920">which</span> <span m="4050430">is</span> <span m="4050600">not</span> <span m="4050820">hard</span> <span m="4051020">to</span> <span m="4051090">prove,</span> <span m="4051430">but</span> <span m="4051540">we</span> <span m="4051640">won't</span> <span m="4051850">do</span> <span m="4052010">it.</span> <span m="4052990">f</span> <span m="4053210">of</span> <span m="4053350">x</span> <span m="4053660">equals</span> <span m="4054090">theta</span> <span m="4054820">of</span> <span m="4054940">g</span> <span m="4055150">of</span> <span m="4055260">x</span> <span m="4057110">if</span> <span m="4057400">and</span> <span m="4057490">only</span> <span m="4059900">f</span> <span m="4060160">of</span> <span m="4060240">the</span> <span m="4060320">x</span> <span m="4061280">equals</span> <span m="4061730">O</span> <span m="4062010">of</span> <span m="4062050">g</span> <span m="4062250">of</span> <span m="4062400">x</span> <span m="4064480">and</span> <span m="4066230">f</span> <span m="4066460">of</span> <span m="4066590">x</span> <span m="4067920">equals</span> <span m="4068510">omega</span> <span m="4068980">g</span> <span m="4069330">of</span> <span m="4069420">x.</span> </p>
<p><span m="4073130">All</span> <span m="4073610">right. So</span> <span m="4073690">let's</span> <span m="4074120">do</span> <span m="4074290">some</span> <span m="4074480">examples</span> <span m="4074935">of</span> <span m="4075390">theta.</span> <span m="4089710">So</span> <span m="4089950">for</span> <span m="4090100">example,</span> <span m="4091620">10x</span> <span m="4092630">cubed</span> <span m="4094120">minus</span> <span m="4094630">20x</span> <span m="4095800">plus</span> <span m="4096100">1</span> <span m="4097490">is</span> <span m="4097729">more</span> <span m="4098300">easily</span> <span m="4098680">written</span> <span m="4098970">as</span> <span m="4099189">theta</span> <span m="4100752">of</span> <span m="4101189">x cubed--</span> <span m="4103529">because</span> <span m="4103740">the</span> <span m="4103819">constants</span> <span m="4104330">don't</span> <span m="4104520">matter.</span> <span m="4104840">The</span> <span m="4104930">low order</span> <span m="4105340">terms</span> <span m="4105680">don't</span> <span m="4105880">matter.</span> </p>
<p><span m="4107319">What</span> <span m="4107439">about</span> <span m="4107689">this?</span> <span m="4108859">Is</span> <span m="4109229">x</span> <span m="4109630">over</span> <span m="4109979">log</span> <span m="4110330">of</span> <span m="4110430">x,</span> <span m="4112850">is</span> <span m="4113040">that</span> <span m="4113359">theta</span> <span m="4113819">of</span> <span m="4113970">x?</span> <span m="4117400">x</span> <span m="4117890">over</span> <span m="4118069">log</span> <span m="4118370">of</span> <span m="4118479">x.</span> <span m="4118760">Is</span> <span m="4118870">that</span> <span m="4119109">theta</span> <span m="4119550">of</span> <span m="4119760">x?</span> <span m="4121500">People</span> <span m="4121770">are</span> <span m="4121859">nodding</span> <span m="4122200">their</span> <span m="4122350">heads.</span> <span m="4124979">Hmm.</span> <span m="4126880">Let's</span> <span m="4127170">see.</span> <span m="4127750">Let's</span> <span m="4128029">check.</span> </p>
<p><span m="4131859">The</span> <span m="4131979">limit</span> <span m="4133050">as</span> <span m="4133300">x</span> <span m="4133590">goes</span> <span m="4133870">to</span> <span m="4133970">infinity</span> <span m="4135340">of</span> <span m="4135899">x</span> <span m="4136330">over</span> <span m="4137229">log</span> <span m="4137560">of</span> <span m="4137680">x</span> <span m="4138830">divided</span> <span m="4139310">by</span> <span m="4139510">x.</span> <span m="4141660">Well, the</span> <span m="4142090">x's</span> <span m="4142490">cancel.</span> <span m="4142920">That's</span> <span m="4143189">the</span> <span m="4143270">limit</span> <span m="4144859">as</span> <span m="4145040">x</span> <span m="4145270">goes</span> <span m="4145529">to</span> <span m="4145640">infinity</span> <span m="4146790">of</span> <span m="4146970">1</span> <span m="4147290">over</span> <span m="4148279">log</span> <span m="4148649">of</span> <span m="4148770">x.</span> <span m="4151240">What's</span> <span m="4151470">that?</span> <span m="4153399">0.</span> <span m="4155460">Uh-oh.</span> <span m="4156640">it</span> <span m="4156810">is</span> <span m="4156990">not</span> <span m="4157450">theta.</span> <span m="4158830">I</span> <span m="4159020">got</span> <span m="4159250">to</span> <span m="4159310">be</span> <span m="4159450">bigger</span> <span m="4159880">than</span> <span m="4160050">zero.</span> <span m="4162300">In</span> <span m="4162460">fact,</span> <span m="4164270">this</span> <span m="4164580">grows</span> <span m="4165050">more</span> <span m="4165300">slowly</span> <span m="4166130">than</span> <span m="4166319">that--</span> <span m="4167880">strictly</span> <span m="4168290">more</span> <span m="4168500">slowly.</span> <span m="4170250">All</span> <span m="4170640">right.</span> <span m="4171930">A</span> <span m="4172029">little</span> <span m="4172310">more</span> <span m="4172490">slowly,</span> <span m="4172840">but</span> <span m="4173080">strictly</span> <span m="4173470">more</span> <span m="4173680">slowly.</span> </p>
<p><span m="4176149">All</span> <span m="4176229">right.</span> <span m="4176520">If</span> <span m="4176660">I</span> <span m="4176729">use</span> <span m="4178310">for</span> <span m="4178770">an</span> <span m="4178930">algorithm,</span> <span m="4179490">if</span> <span m="4179620">you</span> <span m="4179729">want</span> <span m="4179920">to</span> <span m="4179990">say</span> <span m="4180260">the</span> <span m="4180399">algorithm</span> <span m="4180850">runs</span> <span m="4182130">in</span> <span m="4182410">quadratic</span> <span m="4183000">time--</span> <span m="4183359">namely</span> <span m="4183680">at</span> <span m="4183790">least</span> <span m="4184080">quadratic</span> <span m="4184569">time,</span> <span m="4184830">and at</span> <span m="4184979">most</span> <span m="4185210">quadratic</span> <span m="4185660">time--</span> <span m="4185960">then</span> <span m="4186100">you'd</span> <span m="4186229">say</span> <span m="4187080">t</span> <span m="4187330">of</span> <span m="4187450">n</span> <span m="4187649">is</span> <span m="4188010">theta</span> <span m="4188529">of</span> <span m="4188689">n</span> <span m="4188830">squared.</span> <span m="4191166">So let</span> <span m="4191580">me write</span> <span m="4191810">that.</span> <span m="4195190">So</span> <span m="4195420">t</span> <span m="4195690">of</span> <span m="4195810">n</span> <span m="4196240">equals</span> <span m="4196700">theta</span> <span m="4197660">n</span> <span m="4197860">squared</span> <span m="4199940">means</span> <span m="4201950">t</span> <span m="4202390">grows</span> <span m="4204560">quadratically</span> <span m="4209240">in</span> <span m="4209600">n--</span> <span m="4211320">both</span> <span m="4211650">upper</span> <span m="4211890">and</span> <span m="4211980">lower</span> <span m="4212170">bound.</span> </p>
<p><span m="4214094">OK?</span> <span m="4215056">So</span> <span m="4221270">for</span> <span m="4221420">these</span> <span m="4221710">three</span> <span m="4221930">guys,</span> <span m="4222370">to</span> <span m="4222460">summarize--</span> <span m="4227350">sort of</span> <span m="4227550">you</span> <span m="4227690">can</span> <span m="4227790">keep</span> <span m="4228010">track</span> <span m="4228370">of</span> <span m="4228430">them</span> <span m="4228550">pretty</span> <span m="4228760">simply--</span> <span m="4235110">O</span> <span m="4236820">means</span> <span m="4238540">less</span> <span m="4238880">than</span> <span m="4239070">or</span> <span m="4239190">equal.</span> <span m="4240030">Omega</span> <span m="4240670">means</span> <span m="4241510">greater</span> <span m="4241960">than</span> <span m="4242090">or</span> <span m="4242200">equal.</span> <span m="4243572">Theta</span> <span m="4244020">means</span> <span m="4244950">equal</span> <span m="4245990">up</span> <span m="4246210">to</span> <span m="4246270">constant</span> <span m="4246740">factors.</span> </p>
<p><span m="4255870">Now</span> <span m="4256150">there's</span> <span m="4256390">two</span> <span m="4256620">more</span> <span m="4256870">symbols</span> <span m="4257430">here.</span> <span m="4259000">And</span> <span m="4259410">they</span> <span m="4259790">correspond</span> <span m="4260400">very</span> <span m="4260720">naturally</span> <span m="4261270">to</span> <span m="4261420">this</span> <span m="4262850">set-up.</span> <span m="4263760">What</span> <span m="4264010">two</span> <span m="4264170">symbols</span> <span m="4264610">are</span> <span m="4264660">missing</span> <span m="4265060">from</span> <span m="4265230">here</span> <span m="4265500">that</span> <span m="4265620">would</span> <span m="4265760">you</span> <span m="4265830">naturally</span> <span m="4266280">add</span> <span m="4266480">to</span> <span m="4266550">here?</span> <span m="4268900">Yeah.</span> <span m="4269150">Less</span> <span m="4269400">than</span> <span m="4269640">and</span> <span m="4269690">greater</span> <span m="4269990">than.</span> </p>
<p><span m="4271130">So</span> <span m="4271600">less</span> <span m="4272030">than</span> <span m="4273390">is</span> <span m="4273740">little</span> <span m="4274130">o.</span> <span m="4275620">Greater</span> <span m="4276080">than</span> <span m="4276850">is</span> <span m="4277010">little</span> <span m="4277560">omega.</span> <span m="4279430">And</span> <span m="4279770">less</span> <span m="4280020">than</span> <span m="4280250">means</span> <span m="4280780">it's</span> <span m="4280980">less</span> <span m="4281160">than</span> <span m="4281310">or</span> <span m="4281410">equal,</span> <span m="4281720">but</span> <span m="4281890">not</span> <span m="4283340">equal.</span> <span m="4284710">And</span> <span m="4284930">this</span> <span m="4285120">means</span> <span m="4285430">greater</span> <span m="4285760">than</span> <span m="4285850">or</span> <span m="4285910">equal,</span> <span m="4286600">not</span> <span m="4287010">equal.</span> </p>
<p><span m="4288600">And</span> <span m="4288860">let me</span> <span m="4288970">put</span> <span m="4289190">them</span> <span m="4289290">up</span> <span m="4289420">over</span> <span m="4289610">here</span> <span m="4289880">and</span> <span m="4289980">give</span> <span m="4290090">you</span> <span m="4290150">the</span> <span m="4290230">formal</span> <span m="4290610">definitions.</span> <span m="4294854">Eraser.</span> <span m="4312310">OK.</span> <span m="4312740">So</span> <span m="4314230">little</span> <span m="4314620">o.</span> <span m="4320770">as</span> <span m="4320900">we</span> <span m="4320980">have</span> <span m="4321300">f</span> <span m="4321530">of</span> <span m="4321670">x</span> <span m="4322480">equals</span> <span m="4322960">little</span> <span m="4323300">o</span> <span m="4324160">of g</span> <span m="4324280">x</span> <span m="4326690">if</span> <span m="4327150">the</span> <span m="4327260">limit</span> <span m="4330880">of</span> <span m="4331680">f</span> <span m="4331920">of</span> <span m="4332040">x</span> <span m="4332580">over</span> <span m="4332860">g</span> <span m="4333090">of</span> <span m="4333210">x</span> <span m="4336390">equals</span> <span m="4336870">0.</span> <span m="4338310">So</span> <span m="4338590">f</span> <span m="4338830">is</span> <span m="4338960">growing</span> <span m="4339350">strictly</span> <span m="4339940">smaller</span> <span m="4340420">than</span> <span m="4340580">g, so</span> <span m="4341040">that</span> <span m="4341180">when</span> <span m="4341280">I</span> <span m="4341340">take</span> <span m="4341590">the</span> <span m="4341680">ratio,</span> <span m="4342490">the</span> <span m="4342610">limit</span> <span m="4342890">goes</span> <span m="4343110">to</span> <span m="4343190">0.</span> </p>
<p><span m="4344680">And</span> <span m="4344930">the</span> <span m="4345000">reverse</span> <span m="4345500">is</span> <span m="4345760">little</span> <span m="4345990">omega.</span> <span m="4350860">And</span> <span m="4351010">we</span> <span m="4351090">say</span> <span m="4351310">that</span> <span m="4351480">f</span> <span m="4351720">of</span> <span m="4351850">x</span> <span m="4352830">is</span> <span m="4353360">little</span> <span m="4353730">omega--</span> <span m="4355082">whoops--</span> <span m="4355880">of</span> <span m="4356010">g</span> <span m="4356190">of</span> <span m="4356300">x</span> <span m="4358750">if</span> <span m="4359470">this</span> <span m="4360000">is</span> <span m="4360500">infinity.</span> <span m="4362770">Because</span> <span m="4363270">now</span> <span m="4363720">f</span> <span m="4364010">is</span> <span m="4364120">growing</span> <span m="4364450">strictly</span> <span m="4365050">faster</span> <span m="4365470">than</span> <span m="4365610">g,</span> <span m="4366430">so</span> <span m="4366590">I</span> <span m="4366640">take</span> <span m="4366850">the</span> <span m="4366920">limit</span> <span m="4367120">of</span> <span m="4367180">the</span> <span m="4367260">ratio,</span> <span m="4368130">and</span> <span m="4368170">it</span> <span m="4368240">goes</span> <span m="4368450">to</span> <span m="4368550">infinity.</span> </p>
<p><span m="4371110">All</span> <span m="4371290">right</span> <span m="4371560">so</span> <span m="4371670">we</span> <span m="4371740">can</span> <span m="4371880">do</span> <span m="4372070">some</span> <span m="4372260">simple</span> <span m="4372610">examples.</span> <span m="4375510">Then</span> <span m="4375680">we're</span> <span m="4375810">almost</span> <span m="4376200">done</span> <span m="4376380">with</span> <span m="4376650">all</span> <span m="4376840">this</span> <span m="4377030">notation.</span> </p>
<p><span m="4389520">What</span> <span m="4389690">about</span> <span m="4389930">this</span> <span m="4390160">one?</span> <span m="4390680">x</span> <span m="4391140">over</span> <span m="4391450">ln</span> <span m="4391760">of</span> <span m="4391870">x--</span> <span m="4392150">we</span> <span m="4392240">just</span> <span m="4392450">looked</span> <span m="4392660">at</span> <span m="4392810">it--</span> <span m="4394070">and</span> <span m="4395060">x.</span> <span m="4396709">What symbol</span> <span m="4397198">do I put</span> <span m="4397687">there?</span> <span m="4401600">What</span> <span m="4401810">symbol's</span> <span m="4402250">missing</span> <span m="4402600">here?</span> <span m="4404870">Little</span> <span m="4405390">o.</span> <span m="4406060">This</span> <span m="4406510">grows</span> <span m="4406590">strictly</span> <span m="4407030">smaller</span> <span m="4407400">than</span> <span m="4407530">that.</span> <span m="4407890">If</span> <span m="4408010">I</span> <span m="4408100">took</span> <span m="4408360">the</span> <span m="4409050">ratio,</span> <span m="4409730">and</span> <span m="4409830">I</span> <span m="4409870">took</span> <span m="4410040">the</span> <span m="4410110">limit,</span> <span m="4410490">we</span> <span m="4410580">got</span> <span m="4410760">0.</span> <span m="4411390">So</span> <span m="4411570">that's</span> <span m="4411760">a</span> <span m="4411820">little</span> <span m="4412000">o.</span> <span m="4413510">OK.</span> </p>
<p><span m="4415810">Is</span> <span m="4416010">this</span> <span m="4416200">true?</span> <span m="4417340">x</span> <span m="4417650">over</span> <span m="4417880">100--</span> <span m="4418990">is</span> <span m="4419150">that</span> <span m="4419410">little</span> <span m="4419700">o</span> <span m="4419890">of</span> <span m="4420000">x?</span> <span m="4424050">No.</span> <span m="4426410">The</span> <span m="4426550">limit</span> <span m="4426890">of</span> <span m="4426960">the</span> <span m="4427060">ratio</span> <span m="4427520">is</span> <span m="4427620">1</span> <span m="4427830">over</span> <span m="4428040">100.</span> <span m="4428520">That's</span> <span m="4428720">bigger</span> <span m="4428960">than</span> <span m="4429110">0.</span> <span m="4429590">Constants</span> <span m="4430200">don't</span> <span m="4430430">matter.</span> <span m="4431420">All</span> <span m="4431570">right.</span> <span m="4431770">So</span> <span m="4431950">what</span> <span m="4432140">symbol</span> <span m="4432480">would</span> <span m="4432660">go</span> <span m="4432850">here?</span> <span m="4434530">Theta</span> <span m="4435020">goes</span> <span m="4435370">here.</span> <span m="4436060">Good.</span> <span m="4436960">Also</span> <span m="4437360">big</span> <span m="4437600">O</span> <span m="4437705">and</span> <span m="4437810">big</span> <span m="4437950">omega.</span> <span m="4438310">But</span> <span m="4438440">theta</span> <span m="4438950">captures</span> <span m="4439430">everything.</span> </p>
<p><span m="4441640">All right.</span> <span m="4441810">What</span> <span m="4442050">symbol</span> <span m="4442590">goes</span> <span m="4442930">here?</span> <span m="4443990">x</span> <span m="4444350">squared</span> <span m="4444740">equals</span> <span m="4445210">something</span> <span m="4446700">x.</span> <span m="4448330">What</span> <span m="4448500">symbol</span> <span m="4448820">goes</span> <span m="4449040">there?</span> <span m="4451660">Little</span> <span m="4452040">omega.</span> <span m="4452800">It would</span> <span m="4453020">also</span> <span m="4453230">be</span> <span m="4453300">true</span> <span m="4453450">to</span> <span m="4453530">put</span> <span m="4453680">big</span> <span m="4453900">omega,</span> <span m="4454260">but</span> <span m="4454420">little</span> <span m="4454650">omega</span> <span m="4454970">tells</span> <span m="4455290">you</span> <span m="4455390">more.</span> <span m="4456320">Because</span> <span m="4456660">the</span> <span m="4456730">limit</span> <span m="4457020">of</span> <span m="4457120">x</span> <span m="4457260">squared</span> <span m="4457520">over</span> <span m="4457730">x</span> <span m="4458260">goes</span> <span m="4458670">to</span> <span m="4458790">infinity.</span> </p>
<p><span m="4461190">OK.</span> <span m="4461520">Any</span> <span m="4461740">questions</span> <span m="4462320">about</span> <span m="4462590">these</span> <span m="4462790">things?</span> <span m="4464760">So</span> <span m="4465360">a</span> <span m="4465600">natural</span> <span m="4465920">thing</span> <span m="4466080">on</span> <span m="4466150">the</span> <span m="4466220">midterm</span> <span m="4466780">is</span> <span m="4467800">we</span> <span m="4467950">do</span> <span m="4468110">some</span> <span m="4468300">of</span> <span m="4468370">that,</span> <span m="4468690">for</span> <span m="4468800">example.</span> <span m="4469710">All</span> <span m="4470110">right?</span> <span m="4470410">And you'll</span> <span m="4470580">certainly</span> <span m="4470880">see</span> <span m="4471040">some</span> <span m="4471210">of</span> <span m="4471270">that</span> <span m="4471420">on</span> <span m="4471530">homework.</span> <span m="4473040">You</span> <span m="4473190">might</span> <span m="4473420">see</span> <span m="4473540">some</span> <span m="4473690">of</span> <span m="4473760">that</span> <span m="4473940">on</span> <span m="4474060">the</span> <span m="4474130">test.</span> <span m="4475520">You'll</span> <span m="4475700">certainly</span> <span m="4476100">see</span> <span m="4476250">all</span> <span m="4476380">this</span> <span m="4476540">stuff--</span> <span m="4477070">little</span> <span m="4477300">omega</span> <span m="4477690">not</span> <span m="4477945">very</span> <span m="4478200">often</span> <span m="4478760">in</span> <span m="4478960">6046</span> <span m="4479940">and</span> <span m="4480040">later</span> <span m="4480250">algorithms</span> <span m="4480590">courses.</span> <span m="4481020">But</span> <span m="4481030">all</span> <span m="4481180">the</span> <span m="4481270">rest</span> <span m="4481550">you'll</span> <span m="4481660">see</span> <span m="4481860">there.</span> <span m="4483970">Any</span> <span m="4484130">questions?</span> </p>
<p><span m="4486230">All</span> <span m="4486430">right.</span> <span m="4486860">Now</span> <span m="4487720">I'm</span> <span m="4487870">going</span> <span m="4487990">to</span> <span m="4488070">show</span> <span m="4488350">you</span> <span m="4488490">something</span> <span m="4489760">that</span> <span m="4489970">is</span> <span m="4490090">sort</span> <span m="4490330">of</span> <span m="4490400">scary.</span> <span m="4492020">Yeah.</span> </p>
<p><span m="4492924">AUDIENCE: Why</span> <span m="4493376">do you say that</span> <span m="4493828">it's equal to</span> <span m="4494280">zero</span> <span m="4494732">when you</span> <span m="4495190">said it's less</span> <span m="4495671">than</span> <span m="4496633">[INAUDIBLE]</span> <span m="4500000">equals sign.</span> </p>
<p><span m="4501450">PROFESSOR: Which</span> <span m="4501860">equals?</span> </p>
<p><span m="4502820">AUDIENCE: You said</span> <span m="4503250">little omega</span> <span m="4503716">is less</span> <span m="4504182">than.</span> </p>
<p><span m="4505580">PROFESSOR: Little</span> <span m="4505900">omega,</span> <span m="4506730">it</span> <span m="4506980">corresponds</span> <span m="4507590">to</span> <span m="4507680">a</span> <span m="4507810">greater</span> <span m="4508296">than,</span> <span m="4509270">which</span> <span m="4509530">means</span> <span m="4509850">a greater</span> <span m="4509960">than or</span> <span m="4510480">equal</span> <span m="4510870">to,</span> <span m="4511650">but</span> <span m="4511870">not</span> <span m="4512220">an</span> <span m="4512600">equal to.</span> <span m="4514840">And</span> <span m="4515140">the way we</span> <span m="4515340">can</span> <span m="4515460">see</span> <span m="4515720">that</span> <span m="4516220">is</span> <span m="4516670">looking</span> <span m="4516910">at</span> <span m="4516990">the</span> <span m="4517300">definitions</span> <span m="4517510">here</span> <span m="4521010">in</span> <span m="4522200">the</span> <span m="4522300">greater</span> <span m="4522710">than</span> <span m="4522890">or</span> <span m="4523010">equal</span> <span m="4523290">to</span> <span m="4523730">is the</span> <span m="4524080">big</span> <span m="4524340">omega.</span> <span m="4525705">And</span> <span m="4526090">that</span> <span m="4526540">just says</span> <span m="4529010">you're</span> <span m="4529200">bigger</span> <span m="4529500">than</span> <span m="4529660">0.</span> </p>
<p><span m="4532040">Ah.</span> <span m="4532350">Let's</span> <span m="4532540">see.</span> <span m="4533280">All</span> <span m="4533650">right.</span> <span m="4535350">So</span> <span m="4535510">I'm</span> <span m="4535610">bigger</span> <span m="4535870">than</span> <span m="4536030">or</span> <span m="4536110">equal</span> <span m="4536310">to</span> <span m="4536510">is</span> <span m="4536690">omega.</span> <span m="4537680">But</span> <span m="4537820">I'm</span> <span m="4537950">not</span> <span m="4538320">equal</span> <span m="4538740">to,</span> <span m="4539820">which</span> <span m="4540080">is</span> <span m="4540250">theta.</span> <span m="4542970">So</span> <span m="4543190">if</span> <span m="4543280">I</span> <span m="4543390">know</span> <span m="4543790">that</span> <span m="4543920">I'm</span> <span m="4544160">this</span> <span m="4544590">but</span> <span m="4544760">not</span> <span m="4545140">that,</span> <span m="4546940">the</span> <span m="4547100">only</span> <span m="4547270">conclusion</span> <span m="4547830">is</span> <span m="4548120">that</span> <span m="4548250">I'm</span> <span m="4548380">infinity</span> <span m="4549500">in</span> <span m="4549610">the</span> <span m="4549690">ratio.</span> <span m="4551590">All</span> <span m="4551760">right?</span> <span m="4551960">So</span> <span m="4552090">that's</span> <span m="4552310">another</span> <span m="4552510">way</span> <span m="4552680">of</span> <span m="4552830">saying</span> <span m="4553170">it.</span> <span m="4553890">That's</span> <span m="4554140">why.</span> <span m="4555940">Any other</span> <span m="4556431">questions?</span> </p>
<p><span m="4560330">All right.</span> <span m="4560590">Now</span> <span m="4560730">I'm</span> <span m="4560830">going</span> <span m="4560950">to</span> <span m="4561030">show</span> <span m="4561290">you</span> <span m="4561470">something</span> <span m="4561750">that</span> <span m="4561860">is</span> <span m="4562100">commonly</span> <span m="4562690">done,</span> <span m="4563700">and</span> <span m="4563900">that</span> <span m="4564040">many</span> <span m="4564400">students</span> <span m="4564890">will</span> <span m="4565070">attempt</span> <span m="4565420">to</span> <span m="4565490">do</span> <span m="4566000">after</span> <span m="4566380">seeing</span> <span m="4566620">this</span> <span m="4566760">notation,</span> <span m="4567260">because</span> <span m="4567390">the</span> <span m="4567430">notation</span> <span m="4567990">is</span> <span m="4568170">pretty</span> <span m="4568370">simple.</span> <span m="4568720">Take</span> <span m="4568930">your</span> <span m="4569040">limits.</span> <span m="4569390">Everything</span> <span m="4570360">goes</span> <span m="4570590">fine.</span> <span m="4570900">Then</span> <span m="4571000">you</span> <span m="4571070">start</span> <span m="4571440">cruising</span> <span m="4571980">along</span> <span m="4572310">with</span> <span m="4572440">it.</span> </p>
<p><span m="4573290">So</span> <span m="4573460">now</span> <span m="4573660">I'm</span> <span m="4573780">going</span> <span m="4573910">to</span> <span m="4573980">show</span> <span m="4574300">you</span> <span m="4574590">how</span> <span m="4574740">to</span> <span m="4574860">prove</span> <span m="4575200">something</span> <span m="4575490">that's</span> <span m="4575670">false</span> <span m="4577100">using</span> <span m="4577450">this</span> <span m="4577600">notation</span> <span m="4578200">in</span> <span m="4578310">a</span> <span m="4578370">way</span> <span m="4578630">that</span> <span m="4579230">people</span> <span m="4580200">often</span> <span m="4580680">do.</span> <span m="4581770">And</span> <span m="4581890">the</span> <span m="4581980">hope</span> <span m="4582260">is</span> <span m="4582400">by</span> <span m="4582540">showing</span> <span m="4583000">you,</span> <span m="4583792">you</span> <span m="4584190">won't</span> <span m="4584470">do</span> <span m="4584670">it.</span> <span m="4586700">It</span> <span m="4586880">will</span> <span m="4587160">confuse</span> <span m="4587760">you,</span> <span m="4589130">which</span> <span m="4589350">is</span> <span m="4589450">why</span> <span m="4590190">sometimes</span> <span m="4590940">we</span> <span m="4591090">worry</span> <span m="4591360">about</span> <span m="4591910">showing</span> <span m="4592310">you.</span> <span m="4592480">But</span> <span m="4597770">let</span> <span m="4598790">f</span> <span m="4598880">of</span> <span m="4599390">i--</span> <span m="4600620">f of</span> <span m="4600970">n--</span> <span m="4602900">equal</span> <span m="4603210">the</span> <span m="4603300">sum</span> <span m="4604010">i</span> <span m="4604360">equals</span> <span m="4604760">1</span> <span m="4605030">to</span> <span m="4605110">n</span> <span m="4606270">of</span> <span m="4606490">i.</span> </p>
<p><span m="4608420">Now</span> <span m="4608580">we</span> <span m="4608690">all</span> <span m="4608850">know</span> <span m="4609000">what</span> <span m="4609150">that</span> <span m="4609340">is,</span> <span m="4609590">right?</span> <span m="4611600">All</span> <span m="4611810">right.</span> <span m="4612700">So</span> <span m="4612990">I'm</span> <span m="4613080">going</span> <span m="4613210">to</span> <span m="4613280">prove</span> <span m="4613660">to</span> <span m="4613800">you</span> <span m="4616800">that</span> <span m="4617100">f</span> <span m="4617290">of</span> <span m="4617380">n</span> <span m="4618530">is</span> <span m="4618740">O</span> <span m="4618900">of</span> <span m="4618970">n.</span> <span m="4622310">Now what</span> <span m="4622540">really--</span> <span m="4623410">what</span> <span m="4623870">should</span> <span m="4624130">be</span> <span m="4624260">here</span> <span m="4624410">instead</span> <span m="4624700">of</span> <span m="4624820">n?</span> <span m="4626460">n</span> <span m="4626730">squared.</span> <span m="4627980">So</span> <span m="4628720">n</span> <span m="4628920">squared</span> <span m="4629240">can't</span> <span m="4629620">be</span> <span m="4629750">O of</span> <span m="4629920">n.</span> <span m="4630090">So</span> <span m="4630260">this</span> <span m="4630430">can't</span> <span m="4630770">be</span> <span m="4630860">true.</span> </p>
<p><span m="4632020">All</span> <span m="4632260">right.</span> <span m="4632500">Well</span> <span m="4633940">let's</span> <span m="4634150">see</span> <span m="4634500">how</span> <span m="4634630">we</span> <span m="4634730">do</span> <span m="4634930">it.</span> <span m="4637770">All right.</span> <span m="4638070">So</span> <span m="4638140">the</span> <span m="4638200">false</span> <span m="4638610">proof</span> <span m="4642730">is</span> <span m="4642950">by</span> <span m="4643220">induction</span> <span m="4648380">on</span> <span m="4648670">n.</span> <span m="4651280">My</span> <span m="4651450">induction</span> <span m="4651850">hypothesis</span> <span m="4654780">is</span> <span m="4655150">p</span> <span m="4655410">of</span> <span m="4655530">n</span> <span m="4656630">is</span> <span m="4656780">going</span> <span m="4656990">to</span> <span m="4657060">be</span> <span m="4657320">that</span> <span m="4657600">f</span> <span m="4657840">of</span> <span m="4657930">n</span> <span m="4659820">is</span> <span m="4660010">O of</span> <span m="4660190">n.</span> <span m="4662100">All right?</span> <span m="4662410">That's</span> <span m="4662860">what</span> <span m="4662990">we,</span> <span m="4663440">when we</span> <span m="4663530">want</span> <span m="4663690">to</span> <span m="4663760">prove</span> <span m="4664060">something</span> <span m="4664270">by</span> <span m="4664420">induction,</span> <span m="4665020">typically</span> <span m="4665410">that's</span> <span m="4665610">what</span> <span m="4665750">you</span> <span m="4665830">do.</span> <span m="4666480">Becomes the</span> <span m="4666900">induction</span> <span m="4667290">hypothesis.</span> </p>
<p><span m="4669390">Let's</span> <span m="4669510">look</span> <span m="4669640">at</span> <span m="4669710">the</span> <span m="4669780">base</span> <span m="4670050">case.</span> <span m="4674690">Well</span> <span m="4675210">f</span> <span m="4675450">of</span> <span m="4675570">1</span> <span m="4677140">is</span> <span m="4677460">1,</span> <span m="4679670">and</span> <span m="4679890">1</span> <span m="4680240">is</span> <span m="4680440">O</span> <span m="4680630">of</span> <span m="4680730">1</span> <span m="4681020">surely.</span> <span m="4683012">Right?</span> <span m="4683850">Even</span> <span m="4684100">10</span> <span m="4684380">was</span> <span m="4684550">O</span> <span m="4684670">of</span> <span m="4684780">1.</span> <span m="4687006">OK.</span> </p>
<p><span m="4699610">All</span> <span m="4699790">right.</span> <span m="4699920">Now</span> <span m="4700120">we</span> <span m="4700210">do</span> <span m="4700380">the</span> <span m="4700540">inductive</span> <span m="4700980">step.</span> <span m="4708030">We</span> <span m="4708190">assume</span> <span m="4711140">p</span> <span m="4711590">n is true</span> <span m="4713270">to</span> <span m="4713450">prove</span> <span m="4715220">that</span> <span m="4715370">p n</span> <span m="4715580">plus</span> <span m="4715820">1</span> <span m="4716010">is</span> <span m="4716120">true.</span> <span m="4719260">Well</span> <span m="4719370">p</span> <span m="4719770">n</span> <span m="4721030">means</span> <span m="4721620">that</span> <span m="4722890">f of</span> <span m="4723260">n</span> <span m="4724600">is</span> <span m="4724810">O</span> <span m="4724980">of</span> <span m="4725040">n.</span> </p>
<p><span m="4726850">Now</span> <span m="4727130">I</span> <span m="4727220">look</span> <span m="4727470">at</span> <span m="4727530">f</span> <span m="4727690">of</span> <span m="4727810">n</span> <span m="4727950">plus</span> <span m="4728220">1.</span> <span m="4729980">That</span> <span m="4730220">equals</span> <span m="4731700">the</span> <span m="4731810">sum</span> <span m="4732100">of</span> <span m="4732160">the</span> <span m="4732220">first</span> <span m="4732560">n</span> <span m="4732790">plus</span> <span m="4733080">1</span> <span m="4733310">number.</span> <span m="4733630">So</span> <span m="4733760">that</span> <span m="4733870">will</span> <span m="4733960">be</span> <span m="4734550">f</span> <span m="4734860">of n</span> <span m="4735680">plus</span> <span m="4736540">n</span> <span m="4736730">plus</span> <span m="4737000">1.</span> <span m="4739640">The</span> <span m="4739750">induction</span> <span m="4740130">hypothesis</span> <span m="4740890">says</span> <span m="4741360">this</span> <span m="4741640">is</span> <span m="4741800">O of</span> <span m="4742110">n.</span> <span m="4744000">Clearly</span> <span m="4744580">this</span> <span m="4744800">is</span> <span m="4744940">O</span> <span m="4745080">of</span> <span m="4745170">n.</span> </p>
<p><span m="4748720">Well</span> <span m="4749380">I</span> <span m="4749500">got</span> <span m="4749720">O</span> <span m="4749890">of</span> <span m="4749960">n</span> <span m="4750130">plus</span> <span m="4750390">O</span> <span m="4750590">of n.</span> <span m="4750830">That's</span> <span m="4751040">at</span> <span m="4751120">most</span> <span m="4751390">twice</span> <span m="4751890">O of</span> <span m="4752090">n.</span> <span m="4756490">And</span> <span m="4756680">constants</span> <span m="4757180">don't</span> <span m="4757390">matter.</span> <span m="4758630">So</span> <span m="4758810">that's</span> <span m="4759247">O of</span> <span m="4759684">n.</span> </p>
<p><span m="4766080">All</span> <span m="4766280">right.</span> <span m="4766520">This</span> <span m="4766740">should</span> <span m="4768810">begin</span> <span m="4769110">to</span> <span m="4769180">cause</span> <span m="4769750">panic.</span> <span m="4771680">Is</span> <span m="4771830">induction</span> <span m="4772430">wrong?</span> <span m="4772900">What's</span> <span m="4773190">going</span> <span m="4773420">on</span> <span m="4773520">with</span> <span m="4773580">the</span> <span m="4773650">big</span> <span m="4773890">O?</span> <span m="4775520">You</span> <span m="4775690">will</span> <span m="4775810">see</span> <span m="4776080">this</span> <span m="4776360">done</span> <span m="4776610">all</span> <span m="4776900">the</span> <span m="4777000">time.</span> <span m="4777350">People</span> <span m="4778280">would</span> <span m="4778440">do</span> <span m="4778550">this</span> <span m="4778710">kind</span> <span m="4778880">of</span> <span m="4778940">an</span> <span m="4779010">argument.</span> <span m="4780990">Looks</span> <span m="4781240">pretty</span> <span m="4781430">good,</span> <span m="4781700">right?</span> <span m="4784430">Can</span> <span m="4784890">anybody</span> <span m="4785340">see</span> <span m="4785640">where</span> <span m="4787650">I</span> <span m="4787760">got</span> <span m="4787940">off</span> <span m="4788080">track?</span> <span m="4788400">This</span> <span m="4788540">is</span> <span m="4788630">a</span> <span m="4788710">really</span> <span m="4789040">hard</span> <span m="4789300">one</span> <span m="4789440">to</span> <span m="4789500">figure</span> <span m="4789780">out</span> <span m="4789980">where</span> <span m="4790120">things</span> <span m="4790420">went</span> <span m="4790590">wrong.</span> <span m="4791310">Yeah?</span> </p>
<p><span m="4792254">AUDIENCE: One of them might be that</span> <span m="4793670">you might</span> <span m="4794142">be applying</span> <span m="4794614">O</span> <span m="4795090">of n</span> <span m="4796066">to</span> <span m="4796554">numbers</span> <span m="4798018">which</span> <span m="4798994">really</span> <span m="4799970">you</span> <span m="4800458">really</span> <span m="4800946">apply</span> <span m="4801922">asymptotic</span> <span m="4802410">notation</span> <span m="4802898">to functions.</span> </p>
<p><span m="4805040">PROFESSOR: Great.</span> <span m="4806390">That's</span> <span m="4806720">a</span> <span m="4806780">great</span> <span m="4807300">observation.</span> <span m="4808060">Asymptotic</span> <span m="4808780">notation</span> <span m="4810265">only</span> <span m="4810760">works</span> <span m="4811000">for</span> <span m="4811090">functions.</span> <span m="4811700">Now</span> <span m="4812750">here</span> <span m="4813890">when</span> <span m="4814070">we</span> <span m="4814180">say</span> <span m="4814340">1</span> <span m="4814620">is</span> <span m="4814760">O</span> <span m="4815010">of 1,</span> <span m="4815460">well,</span> <span m="4816290">you</span> <span m="4816440">could</span> <span m="4816600">think</span> <span m="4816820">of</span> <span m="4816900">it</span> <span m="4816980">as</span> <span m="4817340">f</span> <span m="4817560">of</span> <span m="4817700">x</span> <span m="4817920">being</span> <span m="4818220">1--</span> <span m="4818640">the</span> <span m="4818710">function--</span> <span m="4819600">and</span> <span m="4819720">so</span> <span m="4819880">forth.</span> <span m="4820240">So</span> <span m="4820260">that's</span> <span m="4820500">OK.</span> <span m="4821480">That's</span> <span m="4821750">a</span> <span m="4821800">great</span> <span m="4822120">point.</span> </p>
<p><span m="4822730">Now let's</span> <span m="4822940">figure</span> <span m="4823250">out</span> <span m="4824410">using</span> <span m="4824830">that</span> <span m="4825140">how</span> <span m="4825310">we got in</span> <span m="4825640">trouble.</span> <span m="4827150">We're</span> <span m="4827220">not</span> <span m="4827390">quite</span> <span m="4827660">done</span> <span m="4827890">yet.</span> <span m="4828040">That's</span> <span m="4828230">a</span> <span m="4828280">great</span> <span m="4828630">observation.</span> <span m="4829670">But</span> <span m="4830540">where's</span> <span m="4830850">the</span> <span m="4830930">step</span> <span m="4831280">that's</span> <span m="4831720">wrong?</span> </p>
<p><span m="4835870">Oh,</span> <span m="4836270">no.</span> <span m="4836590">2</span> <span m="4836770">times</span> <span m="4837120">O</span> <span m="4837270">of</span> <span m="4837380">n</span> <span m="4837560">is</span> <span m="4837770">O</span> <span m="4837940">of</span> <span m="4838060">n.</span> <span m="4838750">If</span> <span m="4838970">I</span> <span m="4839130">have</span> <span m="4839250">a</span> <span m="4839320">function</span> <span m="4839770">that's</span> <span m="4840220">f</span> <span m="4840510">is</span> <span m="4840650">O</span> <span m="4840820">of</span> <span m="4840900">g,</span> <span m="4841720">twice</span> <span m="4842190">f</span> <span m="4842460">is</span> <span m="4842620">O of g.</span> <span m="4843190">Yeah?</span> </p>
<p><span m="4844078">AUDIENCE: It's the end</span> <span m="4844522">of n equals</span> <span m="4844966">1</span> <span m="4845410">[INAUDIBLE].</span> </p>
<p><span m="4848040">PROFESSOR: Is</span> <span m="4848240">it?</span> <span m="4848460">So</span> <span m="4848770">what's</span> <span m="4849280">the</span> <span m="4849360">problem</span> <span m="4849660">now?</span> </p>
<p><span m="4850371">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4852830">PROFESSOR: Well</span> <span m="4853095">f of</span> <span m="4853360">n</span> <span m="4853510">plus</span> <span m="4853820">1--</span> <span m="4854990">this</span> <span m="4855210">statement</span> <span m="4855590">is</span> <span m="4855720">true,</span> <span m="4856100">because</span> <span m="4856450">f</span> <span m="4856630">of</span> <span m="4856720">n</span> <span m="4857270">plus</span> <span m="4857530">1</span> <span m="4858790">is</span> <span m="4859090">the</span> <span m="4859170">sum</span> <span m="4859700">of</span> <span m="4859910">i</span> <span m="4860210">equals</span> <span m="4860690">1</span> <span m="4860960">to</span> <span m="4861040">n</span> <span m="4861250">plus</span> <span m="4861560">1</span> <span m="4861920">of i.</span> <span m="4863400">That's</span> <span m="4863600">1</span> <span m="4863910">plus</span> <span m="4864190">2</span> <span m="4864490">plus</span> <span m="4865510">n</span> <span m="4865810">plus</span> <span m="4866030">n</span> <span m="4866190">plus</span> <span m="4866440">1.</span> <span m="4867360">And</span> <span m="4867560">this</span> <span m="4867800">is</span> <span m="4868160">f of n.</span> <span m="4870630">So</span> <span m="4871081">that's OK.</span> <span m="4879010">Yeah.</span> </p>
<p><span m="4879970">AUDIENCE: [INAUDIBLE]</span> <span m="4883810">that function.</span> </p>
<p><span m="4884770">PROFESSOR: Yeah.</span> <span m="4885070">You</span> <span m="4885240">know,</span> <span m="4886560">you're getting</span> <span m="4887000">closer.</span> <span m="4888820">Yeah</span> <span m="4890460">that's</span> <span m="4890640">a</span> <span m="4890680">little</span> <span m="4890950">flaky,</span> <span m="4892280">because</span> <span m="4892450">I've</span> <span m="4892540">just</span> <span m="4892740">got</span> <span m="4892870">a</span> <span m="4892910">single</span> <span m="4893390">value</span> <span m="4894010">saying</span> <span m="4894300">it's</span> <span m="4894450">O</span> <span m="4894570">of</span> <span m="4894680">one.</span> <span m="4894950">But</span> <span m="4895060">we</span> <span m="4895150">sort of</span> <span m="4895360">argued</span> <span m="4895730">before</span> <span m="4896100">that</span> <span m="4896340">10</span> <span m="4896620">was</span> <span m="4896800">O</span> <span m="4896950">of</span> <span m="4897030">1.</span> <span m="4898120">Now</span> <span m="4898270">realize</span> <span m="4898660">a</span> <span m="4898710">function</span> <span m="4899190">behind</span> <span m="4899570">here</span> <span m="4899730">that</span> <span m="4899860">matters.</span> <span m="4900980">So</span> <span m="4901130">you're</span> <span m="4901240">on</span> <span m="4901370">the</span> <span m="4901450">right</span> <span m="4901650">trail.</span> <span m="4901970">But there's</span> <span m="4902110">an</span> <span m="4902180">even</span> <span m="4902940">bigger</span> <span m="4903260">whopper</span> <span m="4903740">hiding.</span> <span m="4904160">Yeah.</span> </p>
<p><span m="4904984">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4906220">PROFESSOR: Oh</span> <span m="4906600">yeah.</span> <span m="4906950">This</span> <span m="4907320">is</span> <span m="4907610">really,</span> <span m="4908330">really</span> <span m="4908580">bad.</span> <span m="4911130">All</span> <span m="4911450">right.</span> <span m="4912520">So</span> <span m="4913060">now</span> <span m="4913370">explain</span> <span m="4913950">to</span> <span m="4914020">me</span> <span m="4914210">why</span> <span m="4914480">this</span> <span m="4914690">is</span> <span m="4914830">really</span> <span m="4915315">bad</span> <span m="4915800">what I did here.</span> </p>
<p><span m="4916770">AUDIENCE: You didn't need</span> <span m="4917255">the function</span> <span m="4917740">[INAUDIBLE].</span> </p>
<p><span m="4920140">PROFESSOR: Yeah.</span> <span m="4920570">And</span> <span m="4920850">why</span> <span m="4921210">isn't--</span> <span m="4921540">this</span> <span m="4921790">looks</span> <span m="4922120">like</span> <span m="4922320">a</span> <span m="4922380">function.</span> <span m="4922840">Why</span> <span m="4923100">isn't</span> <span m="4923300">it</span> <span m="4923430">a</span> <span m="4923500">function?</span> <span m="4927010">It's</span> <span m="4927220">not.</span> <span m="4927740">But</span> <span m="4927850">it</span> <span m="4927960">looks</span> <span m="4928200">like</span> <span m="4928410">one.</span> <span m="4929450">f of</span> <span m="4929810">n sure</span> <span m="4930170">looks</span> <span m="4930410">like</span> <span m="4930600">one.</span> <span m="4930760">Why</span> <span m="4931030">is</span> <span m="4931150">it</span> <span m="4931370">not</span> <span m="4931610">a</span> <span m="4931650">function</span> <span m="4932757">when it's</span> <span m="4933214">sitting in</span> <span m="4933671">this thing?</span> <span m="4936870">This</span> <span m="4937020">P of</span> <span m="4937350">n,</span> <span m="4938930">right?</span> <span m="4939730">Because</span> <span m="4939880">I've</span> <span m="4940030">fixed</span> <span m="4940550">the</span> <span m="4940630">value</span> <span m="4941040">of</span> <span m="4941150">n</span> <span m="4941400">here.</span> <span m="4943350">And</span> <span m="4943440">so</span> <span m="4943590">this</span> <span m="4943950">is</span> <span m="4944110">now</span> <span m="4944370">just</span> <span m="4944590">a</span> <span m="4944630">scalar.</span> </p>
<p><span m="4945870">In</span> <span m="4946030">fact,</span> <span m="4947290">you</span> <span m="4947410">know,</span> <span m="4947790">I</span> <span m="4947890">could</span> <span m="4948090">state</span> <span m="4948270">the</span> <span m="4948350">following--</span> <span m="4948710">f</span> <span m="4949000">of</span> <span m="4950420">a</span> <span m="4950530">million,</span> <span m="4954020">right,</span> <span m="4954480">is</span> <span m="4954700">O</span> <span m="4954820">of</span> <span m="4954910">1,</span> <span m="4955910">because</span> <span m="4956220">this</span> <span m="4956390">is</span> <span m="4956520">really</span> <span m="4956880">some</span> <span m="4957270">function</span> <span m="4958650">H</span> <span m="4958930">of</span> <span m="4959070">x</span> <span m="4959750">that</span> <span m="4959890">always</span> <span m="4960300">is</span> <span m="4960410">this</span> <span m="4960600">value.</span> <span m="4961930">So</span> <span m="4962010">the</span> <span m="4962100">difference</span> <span m="4962450">between</span> <span m="4962630">a</span> <span m="4962680">scalar</span> <span m="4963320">and</span> <span m="4963420">a</span> <span m="4963470">function.</span> <span m="4964470">And</span> <span m="4964700">this</span> <span m="4964900">is</span> <span m="4965000">just</span> <span m="4965340">so</span> <span m="4965650">nasty</span> <span m="4966150">looking,</span> <span m="4966380">because</span> <span m="4966740">we</span> <span m="4967130">see</span> <span m="4967330">f</span> <span m="4967450">of</span> <span m="4967520">n.</span> <span m="4967710">That's</span> <span m="4967910">a</span> <span m="4967950">function.</span> </p>
<p><span m="4969140">But</span> <span m="4969360">you</span> <span m="4969450">can</span> <span m="4969620">never</span> <span m="4970220">use</span> <span m="4971300">big</span> <span m="4971550">O</span> <span m="4972540">in</span> <span m="4973590">a</span> <span m="4973690">predicate,</span> <span m="4975190">all right,</span> <span m="4975470">because</span> <span m="4975620">we've</span> <span m="4975820">lost</span> <span m="4976270">the</span> <span m="4976350">function</span> <span m="4976780">here.</span> <span m="4976990">This</span> <span m="4977210">is</span> <span m="4977320">just</span> <span m="4977760">f</span> <span m="4978100">valued</span> <span m="4978620">at</span> <span m="4978770">n</span> <span m="4979450">period.</span> <span m="4980690">That</span> <span m="4980920">single</span> <span m="4981290">instance</span> <span m="4981560">is</span> <span m="4981830">all</span> <span m="4981970">that's</span> <span m="4982130">happened</span> <span m="4982460">here.</span> <span m="4983430">It's</span> <span m="4983620">not</span> <span m="4983860">a</span> <span m="4983890">function</span> <span m="4984300">anymore.</span> <span m="4985910">All</span> <span m="4986050">right.</span> <span m="4986790">If we</span> <span m="4987250">had</span> <span m="4987370">f</span> <span m="4987620">n</span> <span m="4987770">of</span> <span m="4987890">x</span> <span m="4988200">or</span> <span m="4988280">something,</span> <span m="4988680">that</span> <span m="4988880">might</span> <span m="4989060">be</span> <span m="4989180">OK.</span> <span m="4989480">But</span> <span m="4989610">there's</span> <span m="4989780">no</span> <span m="4989930">function</span> <span m="4990360">left.</span> </p>
<p><span m="4991750">All</span> <span m="4991890">right.</span> <span m="4992040">So</span> <span m="4992220">never</span> <span m="4993050">ever</span> <span m="4994300">use</span> <span m="4994720">asymptotic</span> <span m="4995420">notation</span> <span m="4996100">with</span> <span m="4996260">inductive</span> <span m="4996770">proofs,</span> <span m="4997670">because</span> <span m="4998090">you</span> <span m="4998210">will</span> <span m="4998390">just</span> <span m="4998860">invariably</span> <span m="4999690">stick</span> <span m="5000240">big</span> <span m="5000480">O</span> <span m="5001590">into</span> <span m="5001830">your</span> <span m="5001980">predicate.</span> <span m="5003150">And</span> <span m="5003360">you</span> <span m="5003450">could</span> <span m="5003580">prove</span> <span m="5003780">all</span> <span m="5004000">sorts</span> <span m="5004330">of</span> <span m="5004420">false</span> <span m="5004730">things</span> <span m="5005030">once</span> <span m="5005220">you</span> <span m="5005290">do</span> <span m="5005460">that.</span> <span m="5007190">OK?</span> <span m="5009590">All</span> <span m="5009730">right.</span> <span m="5010150">Never</span> <span m="5010620">use</span> <span m="5010830">asymptotic</span> <span m="5011390">notation</span> <span m="5011910">in</span> <span m="5012020">induction</span> <span m="5012440">or</span> <span m="5012500">predicates.</span> <span m="5013400">Thanks.</span> </p>
</div>
        <div id="vid_related" itemprop="description" class="tabContent hide">
<h2 class="subhead">Free Downloads</h2>
<h3 class="subsubhead">Video</h3>
<ul>
<li>iTunes U (<a href="http://itunes.apple.com/us/itunes-u/lecture-13-sums-asymptotics/id503873536?i=110644972">MP4 - 183MB</a>)</li>
<li>Internet Archive (<a href="http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec13_300k.mp4">MP4 - 183MB</a>)</li>
</ul>
<br>
<h3 class="subsubhead">Caption</h3>
<ul>
<li>English-US (<a href="../../../contents/video-lectures/lecture-13-sums-and-asymptotics/X9eErxRjQEI.srt">SRT</a>)</li>
</ul>
</div>
    
   </div>  




      					 
        <div class="" id="parent-fieldname-bottom_html_area">
            
            
        </div>
    
               </main><!--Course_inner_media tag close -->
           		</div>
<!--Course_wrapper tag close -->
            </div>
<!--left tag close -->
            <aside id="right">
                <!--Begin Right Portion -->
                    <div>
    



</div>

                	<div>
    



</div>


        <div class="" id="parent-fieldname-rsi_top_html_area">
            
            
        </div>
    

<!-- RSI google ad space-->



<div id="google_ads">    
    <script async="async" src="https://www.googletagservices.com/tag/js/gpt.js"></script>
    <script type="text/javascript">var googletag = googletag || {}; googletag.cmd = googletag.cmd || [];</script>
    <script type="text/javascript">
googletag.cmd.push(function() {googletag.defineSlot('/1064917/VIDEO_INDIVIDUAL_SLOT_A_DL', [[300, 250], [300, 300], [180, 200], [180, 150], [160, 600]], 'VIDEO_INDIVIDUAL_SLOT_A_DL').addService(googletag.pubads());googletag.defineSlot('/1064917/VIDEO_INDIVIDUAL_SLOT_B_DL', [[300, 250], [300, 300], [180, 200], [180, 150], [160, 600]], 'VIDEO_INDIVIDUAL_SLOT_B_DL').addService(googletag.pubads());googletag.defineSlot('/1064917/VIDEO_INDIVIDUAL_SLOT_C_DL', [[300, 250], [300, 300], [180, 200], [180, 150], [160, 600]], 'VIDEO_INDIVIDUAL_SLOT_C_DL').addService(googletag.pubads());
googletag.pubads().enableSingleRequest();
 googletag.enableServices();
});</script>
    <script language="javascript" type="text/javascript">
googletag.cmd.push(function() {googletag.pubads().set("TYPE","HOUSE");googletag.pubads().set("DEPARTMENT","6");googletag.pubads().set("CRS_BEG2","04");googletag.pubads().set("CRS_END","2J");googletag.pubads().set("SESSION","F");googletag.pubads().set("YEAR","10");})
</script>
    
    <div id="VIDEO_INDIVIDUAL_SLOT_A_DL">
    	<script>googletag.cmd.push(function() { googletag.display('VIDEO_INDIVIDUAL_SLOT_A_DL'); });</script>
    </div>
    <div id="VIDEO_INDIVIDUAL_SLOT_B_DL">
    	<script>googletag.cmd.push(function() { googletag.display('VIDEO_INDIVIDUAL_SLOT_B_DL'); });</script>
    </div>
    <div id="VIDEO_INDIVIDUAL_SLOT_C_DL">
    	<script>googletag.cmd.push(function() { googletag.display('VIDEO_INDIVIDUAL_SLOT_C_DL'); });</script>
    </div>
</div>

<!-- End RSI ads--> 


<div>
    



</div>

            </aside><!--Right div close -->
            <div class="clear"></div>
        </div>
<!--grid tag close -->
      </div>
		
		<footer id="bottom">
			<div id="grid">
				
<div id="portletwrapper-6f63772e626f74746f6d706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d666f6f746572" class="portletWrapper kssattr-portlethash-6f63772e626f74746f6d706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d666f6f746572">
<div class="portletStaticText portlet-static-site-footer"><div id="footer">
<nav aria-label="Footer">     <nav id="foot-c1" class="grid_2 alpha" aria-labelledby="f-find-courses">       <span class="footer" id="f-find-courses" aria-hidden="true">Find Courses</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/courses/find-by-topic/">Find by Topic</a></li>
    <li><a href="https://ocw.mit.edu/courses/find-by-number/">Find by Course Number</a></li>
    <li><a href="https://ocw.mit.edu/courses/find-by-department/">Find by Department</a></li>
    <li><a href="https://ocw.mit.edu/courses/new-courses/">New Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/most-visited-courses/">Most Visited Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/ocw-scholar/">OCW Scholar Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/audio-video-courses/">Audio/Video Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/online-textbooks/">Online Textbooks</a></li>
    <li><a href="https://ocw.mit.edu/courses/instructor-insights/">Instructor Insights</a></li>
    <li><a href="https://ocw.mit.edu/resources/">Supplemental Resources</a></li>
    <li><a href="https://ocw.mit.edu/courses/mitx-related-courseware/">MITx &amp; Related OCW Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/mit-open-learning-library/">MIT Open Learning Library</a></li>
    <li><a href="https://ocw.mit.edu/courses/translated-courses/">Translated Courses</a></li>
</ul>
</nav>
<div id="foot-c2" class="grid_2"><nav aria-labelledby="f-for-educators">         <span id="f-for-educators" class="footer" aria-hidden="true">For Educators</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://chalk-radio.simplecast.com/">Chalk Radio Podcast</a></li>
    <li><a href="https://ocw.mit.edu/educator/">OCW Educator Portal </a></li>
    <li><a href="https://ocw.mit.edu/courses/instructor-insights/">Instructor Insights by Department</a></li>
    <li><a href="https://openlearning.mit.edu/campus/digital-innovations/" aria-label="External Link: Residential Digital Innovations">Residential Digital Innovations </a></li>
    <li><a href="https://ocw.mit.edu/high-school/">OCW Highlights for High School</a></li>
    <li><a href="https://ocw.mit.edu/educator/additional-resources/">Additional Resources</a></li>
</ul>
</nav></div>
<nav class="grid_2" id="foot-c3" aria-labelledby="f-donate">       <span id="f-donate" class="footer" aria-hidden="true">Give Now</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/give/">Make a Donation</a></li>
    <li><a href="https://ocw.mit.edu/give/why-give/">Why Give?</a></li>
    <li><a href="https://ocw.mit.edu/give/our-supporters/">Our Supporters</a></li>
    <li><a href="https://ocw.mit.edu/give/other-ways-to-contribute/">Other Ways to Contribute</a></li>
    <li><a href="https://ocw.mit.edu/support/">Become a Corporate Sponsor</a></li>
</ul>
</nav>
<div class="grid_2" id="foot-c4">
<nav aria-labelledby="f-about">         <span id="f-about" class="footer" aria-hidden="true">About</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/about/">About OpenCourseWare</a></li>
    <li><a href="https://ocw.mit.edu/about/site-statistics/">Site Statistics</a></li>
    <li><a href="https://ocw.mit.edu/about/ocw-stories/">OCW Stories</a></li>
    <li><a href="https://ocw.mit.edu/about/newsletter/">Newsletter</a></li>
    <li><a href="https://www.ocw-openmatters.org/">Open Matters Blog</a></li>
</ul>
</nav><!--about-->       <nav aria-labelledby="f-tools">         <span id="f-tools" class="footer" aria-hidden="true">Tools</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/help/">Help &amp; FAQs</a></li>
    <li><a href="https://ocw.mit.edu/about/contactus">Contact Us</a></li>
    <li><a href="https://accessibility.mit.edu/" target="_blank">Accessibility</a></li>
    <li><a href="https://ocw.mit.edu/help/site-map/">Site Map</a></li>
    <li><a href="../../../common/terms/index.htm">Privacy &amp; Terms of Use</a></li>
    <li><a href="https://ocw.mit.edu/help/rss/">RSS Feeds</a></li>
</ul>
</nav><!--tools-->
</div>
</nav> <aside style="min-height: 289px;" aria-labelledby="f-our-corporate-supporters" class="grid_4 omega" id="foot-c5">           <span aria-hidden="true" class="footer" id="f-our-corporate-supporters">Our Corporate Supporters</span>           <!-- HOME_CORP_LOGO_1 -->
<div id="div-gpt-ad-1388181177156-0" class="sponsors_google_ads_even"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-0'); });
            </script></div>
<!-- HOME_CORP_LOGO_2 -->
<div id="div-gpt-ad-1388181177156-1" class="sponsors_google_ads_odd"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-1'); });
            </script></div>
<!-- HOME_CORP_LOGO_3 -->
<div id="div-gpt-ad-1388181177156-2" class="sponsors_google_ads_even"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-2'); });
            </script></div>
<!-- HOME_CORP_LOGO_4 -->
<div id="div-gpt-ad-1388181177156-3" class="sponsors_google_ads_odd"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-3'); });
            </script></div>
<!-- HOME_CORP_LOGO_5 -->
<div id="div-gpt-ad-1388181177156-4" class="sponsors_google_ads_even"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-4'); });
              </script></div>
<!-- HOME_CORP_LOGO_6 -->
<div id="div-gpt-ad-1388181177156-5" class="sponsors_google_ads_odd"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-5'); });
              </script></div>
</aside>
<div class="grid_12 alpha omega" itemprop="publisher" itemscope="" itemtype="http://schema.org/CollegeOrUniversity">
<h4 class="footer" style="border-top: thin solid #d5c9ba; padding-top: 10px; margin-bottom: 10px;">About <span itemprop="name">MIT OpenCourseWare</span>
</h4>
<p style="color: #999; font-size: 1em; line-height: 1.5em; margin-top: 10px;" itemprop="description">MIT OpenCourseWare is an online publication of materials from over 2,500 MIT courses, freely sharing knowledge with learners and educators around the world. <a href="https://ocw.mit.edu/about/">Learn more »</a></p>
</div>
<div id="foot-copy" class="grid_12 alpha omega" style="border-top: none;">
<a href="http://web.mit.edu"><img src="../../../common/images/logo_mit.png" alt="Massachusetts Institute of Technology logo and name." style="width: 195; height: 44;"></a><a href="https://openlearning.mit.edu/"><img src="https://ocw.mit.edu/images/mitol_logo.png" alt="MIT Open Learning logo and name." style="width: 265; height: 50; vertical-align: top; padding-left:30px;"></a><a href="https://www.oeglobal.org/"><img src="https://ocw.mit.edu/images/oeg_logo.gif" alt="Open Education Consortium logo." style="width: 219px; height: 59px; vertical-align: top; padding-left:20px;"></a><a rel="license" itemprop="useRightsUrl" href="http://creativecommons.org/licenses/by-nc-sa/4.0/"><img src="../../../common/images/cc_by-nc-sa.png" alt="Creative Commons logo with terms BY-NC-SA." style="width: 126px; height: 44px; margin-right: 0; padding-left: 20px;"></a>
<p class="copyright">© 2001–2018<br>
Massachusetts Institute of Technology</p>
<p style="font-size: 0.9em; margin-bottom: 15px;">Your use of the MIT OpenCourseWare site and materials is subject to our <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" rel="license">Creative Commons License</a> and other <a href="../../../common/terms/index.htm" rel="cc:morePermissions">terms of use</a>.</p>
</div>
</div></div>

</div>





                
			</div> <!-- bottom grid end -->
		</footer><!-- footer bottom end -->


   </body>
 </html>
